$k$-stretchability of entanglement, and the duality of $k$-separability and $k$-producibility

The notions of $k$-separability and $k$-producibility are useful and expressive tools for the characterization of entanglement in multipartite quantum systems, when a more detailed analysis would be infeasible or simply needless. In this work we reveal a partial duality between them, which is valid also for their correlation counterparts. This duality can be seen from a much wider perspective, when we consider the entanglement and correlation properties which are invariant under the permutations of the subsystems. These properties are labeled by Young diagrams, which we endow with a refinement-like partial order, to build up their classification scheme. This general treatment reveals a new property, which we call $k$-stretchability, being sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated (or separable) from one another.

IV. An alternative way of introducing permutation invariance 9 V. k-partitionability, k-producibility and k-stretchability 10 A. k-partitionability and k-producibility of correlation and entanglement 11 B. Duality by conjugation 12 C. k-stretchability of correlation and entanglement 13 D. Relations among k-partitionability, k-producibility and k-stretchability The investigation of the correlations among the parts of a composite physical system is an essential tool in statistical physics. If the system is described by quantum mechanics, then nonclassical forms of correlations arise, the most notable is entanglement [1][2][3][4]. It is the main resource of quantum information theory [5][6][7], and its nonclassical properties, playing important role also in many-body physics [8][9][10], make it influential in the behavior and characterization of strongly correlated systems [11][12][13][14].
Even the partial correlation and partial entanglement properties are getting too involved rapidly, with the increasing of the number of subsystems. Singling out particularly expressive properties, we consider the kpartitionability and k-producibility of correlation and of entanglement. For n subsystems, k ranges from 1 to n, so the number of these properties scales linearly with the number of subsystems, moreover, their structures are the simplest possible ones, chains. (In the case of entanglement, k-partitionability is called k-separability [35,37], while k-producibility [46][47][48][49] is also called entanglement depth [50][51][52]. Here we use both concepts for correlation and also for entanglement, this is why we use the naming "k-partitionability of correlation" and "k-producibility of correlation", "k-partitionability of entanglement" and "k-producibility of entanglement" [14,40,41].) These characterize the strength of two different (one-parameter-) aspects of multipartite correlation and entanglement: those which cannot be restricted inside at least k parts, and those which cannot be restricted inside parts of size at most k, respectively. k-producibility plays particularly important role in quantum metrology [53]. k-producibly entangled states for larger k lead to higher sensitivity, so better precision in phase estimation, which has been illustrated in experiments [54][55][56], and which also leads to k-producibility entanglement criteria [57].
k-partitionability and k-producibility are special cases of properties invariant under the permutation of the subsystems. The permutation invariant correlation properties are based on the integer partitions [45,58], also known (represented) as Young diagrams. For the description of the structure of these, we introduce a new order over the integer partitions, called refinement, induced by the refinement order over set partitions, used for the description of the structure of the partial correlation or entanglement properties [40]. The structure of the permutation invariant properties contains the chains of k-partitionability and k-producibility, and it is simpler than the structure describing all the properties. The number of these scales still rapidly with the number of subsystems, but slower than the number of all the partial correlation or entanglement properties.
The general treatment of the permutation invariant properties reveals a partial duality between kpartitionability and k-producibility, which is the manifestation of a duality on a deeper level, relating important properties of Young diagrams, by which kpartitionability and k-producibility are formulated.
The general treatment of the permutation invariant properties reveals also a particularly expressive new property, which we call k-stretchability, leading to the definitions of "k-stretchability of correlation" and "kstretchability of entanglement". It combines the advantages of k-partitionability and k-producibility. Namely, k-partitionability is about the number of subsystems uncorrelated (or separable) with one another, and not sensitive to the size of correlated (or entangled) subsystems [47]; and k-producibility is about the size of the largest correlated (or entangled) subsystem, and not sensitive to the number of subsystems uncorrelated (or separable) from one another; while k-stretchability is sensitive to both of these, in a balanced way. The price to pay for this is that we have roughly two times more k-stretchability properties, k goes from −(n − 1) to n − 1. However, k-stretchability is linearly ordered with k, while the relations between k-partitionability and k-producibility are far more complicated.
The organization of this work is as follows. In section II, we recall the structure of multipartite correlation and entanglement. In section III, we construct the parallel structure for the permutation invariant case. In section IV, we show another introduction of the permutation invariant classification, which can although be considered simpler, but less transparent. In section V, we recall k-partitionability and k-producibility, introduce kstretchability, and show how these properties are related to each other. In section VI, summary, remarks and open questions are listed. In appendix A, we work out the "coarsening" step and some other tools, used in the main text. In appendix B, we present the proofs of some further propositions given in the main text.

II. MULTIPARTITE CORRELATION AND ENTANGLEMENT
Here we recall the structure of the classification and quantification of multipartite correlation and entanglement [14,40,41]. Our goal is to do this in the way sufficient to see how the permutation invariant properties can be formulated parallel to this in the next section.

A. Level 0: subsystems
The classification scheme we present here is rather general. The elementary and composite subsystems can be any discrete finite systems possessing probabilistic description, supposed that the joint systems can be represented by the use of tensor products, which is the basic tool in the constructions. Such systems can be distinguishable quantum systems, second quantized bosonic systems, second quantized fermionic systems with fermion number parity superselection rule imposed, or even classical systems, with significant simplification in the structure.
Let L be the set of the labels of |L| = n elementary subsystems. All the (possibly composite) subsystems are then labeled by the subsets X ⊆ L, the set of which, P 0 := 2 L , naturally possesses a Boolean lattice structure with respect to the inclusion ⊆, which is now denoted with . For every elementary subsystem i ∈ L, let the Hilbert space H i be associated with it, where 1 < dim H i < ∞. From these, for every subsystem X ∈ P 0 , the Hilbert space associated with it is H X = i∈X H i . (For the trivial subsystem X = ∅, we have the one-dimensional Hilbert space H ∅ = Span{|ψ } ∼ = C [40].) The states of the subsystems X ∈ P 0 are given by density operators (positive semidefinite operators of trace 1) acting on H X , the sets of those are denoted with D X .
The mixedness of a state X ∈ D X of a subsystem X can be characterized by the von Neumann entropy [7,[59][60][61] S( X ) = − Tr X ln X ; (1a) and the distinguishability of two states X , σ X ∈ D X of subsystem X can be characterized by the Umegaki relative entropy [6,7,[60][61][62][63][64] The von Neumann entropy is monotone increasing in bistochastic quantum channels, and the relative entropy is monotone decreasing in quantum channels [6,61,63] and

B. Level I: set partitions
For handling the different possible splits of a composite system into subsystems, we need to use the mathematical notion of (set) partition [44] of the system L. The partitions of L are sets of subsystems, ξ = {X 1 , X 2 , . . . , X |ξ| }, where the parts, X ∈ ξ, are nonempty disjoint subsystems, for which X∈ξ X = L. The set of the partitions of L is denoted with Its size is given by the Bell numbers [65], which are rapidly growing with n. There is a natural partial order over the partitions, which is called refinement , given as  [44,45]. With respect to the partitions ξ ∈ P I , we can define the partial correlation and entanglement properties, as well as the measures quantifying them. The ξ-uncorrelated states are those which are products with respect to the partition ξ, D ξ-unc := L ∈ D L ∀X ∈ ξ, ∃ X ∈ D X : L = X∈ξ X ; (3a) the others are ξ-correlated states. With respect to the finest partition ξ = ⊥, we call ⊥-uncorrelated states i∈L i simply uncorrelated states. The expectation val-ues of all ξ-local observables factorize if and only if the state is ξ-uncorrelated. The ξ-uncorrelated states are exactly those which can be prepared from uncorrelated states by ξ-local operations. (This is abbreviated as ξ-LO. With respect to the finest partition ξ = ⊥, we write simply LO.) The ξ-separable states are convex combinations (or statistical mixtures) of ξ-uncorrelated states, the others are ξ-entangled states. (Here, and in the entire text, {p j | j = 1, 2, . . . , m} is a probability distribution, p j ≥ 0 and j p j = 1.) The ξ-separable states are exactly those which can be prepared from uncorrelated states by ξ-local operations and classical communication among the parts X ∈ ξ. (This is abbreviated as ξ-LOCC. With respect to the finest partition ⊥, we write simply LOCC.) Another point of view is that ξ-separable states are exactly those which can be prepared from uncorrelated states by mixtures of ξ-LOs. We also have that D ξ-unc is closed under ξ-LO, D ξ-sep is closed under ξ-LOCC [40,41]. Clearly, if a state is product with respect to a partition, then it is also product with respect to any coarser partition, it is always free to forget about some tensor product signs. This means that these properties show the same lattice structure as the partitions [40,41], P I , that is, One can define the corresponding (informationgeometry based) correlation and entanglement measures [14,40] for all ξ-correlation and ξ-entanglement. These are the most natural generalizations of the mutual information [6,7], the entanglement entropy [16], and the entanglement of formation [17] for Level I of the multipartite case.
The ξ-correlation of a state is its distinguishability by the relative entropy (1b) from the ξ-uncorrelated states [4,14,40,66], given in terms of the von Neumann entropies (1a) of the reduced, or marginal states X = Tr L\X of subsystems X ∈ ξ. Pure states of classical systems are always uncorrelated, the correlation in pure states is of quantum origin, this is what we call entanglement [1-3, 14, 15, 40, 41]. So, for pure states, the measure of entanglement should be that of correlation. The ξ-entanglement of a pure state π = |ψ ψ| ∈ D L is its ξ-correlation, and for mixed states, one can use, e.g., the convex roof extension [17,67,68] to define the ξ-entanglement of for- where j p j π j is a pure decomposition of the state [40].
We have that C ξ is a correlation monotone (not increasing w.r.t. ξ-LO, for the proof see appendix B 1), E ξ is a strong entanglement monotone (convex and not increasing on average w.r.t. selective ξ-LOCC [22,23,40]), and both of these are faithful, C ξ ( ) = 0 ⇔ ∈ D ξ-unc , E ξ ( ) = 0 ⇔ ∈ D ξ-sep [40], moreover, they show the same lattice structure as the partitions, P I , that is, which is called multipartite monotonicity [14,40]. Note that, for two subsystems, for the only nontrivial partition we have that E 1|2 (π) = 2S(π 1 ) = 2S(π 2 ) is just two times the usual entanglement entropy. However, we have a way of derivation completely different than the usual, based on asymptotic LOCC convertibility from Bell-pairs [16]. The usual way cannot be generalized to the multipartite scenario (there is no "reference state", which was the Bell-pair in the bipartite scenario), while our correlation, or statistical physical approach above could straightforwardly be generalized to the multipartite scenario not only here, but also for the Level II properties in the next subsystem.

C. Level II: multiple set partitions
In multipartite entanglement theory, it is necessary to handle mixtures of states uncorrelated with respect to different partitions [35,37,40]. For example, there are tripartite states which cannot be written as a mixture of a given kind of, e.g., 12|3-uncorrelated states, that is, not 12|3-separable, while can be written as a mixture of 12|3-uncorrelated and 13|2-uncorrelated states. Such states should not be considered fully tripartite-entangled, since there is no need for genuine tripartite entangled states in the mixture [35,37,39,40]. Also, if a state is 12|3-separable and also 13|2-separable, it is not necessarily 1|2|3-separable. On the other hand, the order isomorphisms (4a)-(4b) tell us that if we consider states uncorrelated (or separable) with respect to a partition, then we automatically consider states uncorrelated (or separable) with respect to all finer partitions.
To embed these requirements in the labeling of the multipartite correlation and entanglement properties, we use the nonempty down-sets (nonempty ideals) of partitions [40], which are ξ = {ξ 1 , ξ 2 , . . . , ξ |ξ| } ⊆ P I sets of partitions which are closed downwards with respect to . That is, if ξ ∈ ξ, then for all partitions υ ξ we have υ ∈ ξ. The set of the nonempty partition ideals is denoted with This possesses a lattice structure with respect to the standard inclusion as partial order which we also call refinement, and also denote with , (For illustrations, see figure 1.) The least upper and greatest lower bounds ∨, ∧ are then the union and the intersection ∪, ∩.
With respect to the partition ideals ξ ∈ P II , we can define the partial of correlation and entanglement properties, as well as the measures quantifying these.
The ξ-uncorrelated states are those which are ξuncorrelated (3a) with respect to a ξ ∈ ξ, the others are ξ-correlated states. The ξ-uncorrelated states are exactly those which can be prepared from uncorrelated states by ξ-LO for a partition ξ ∈ ξ.
The ξ-separable states are convex combinations of ξuncorrelated states, the others are ξ-entangled states. The ξ-separable states are exactly those which can be prepared from uncorrelated states by mixtures of ξ-LOs for different partitions ξ ∈ ξ. (Note that such transformations do not form a semigroup.) We also have that D ξ-unc is closed under LO, D ξ-sep is closed under LOCC [40,41]. It follows from (4) that these properties show the same lattice structure as the partition ideals [40], P II , that is, One can define the corresponding (informationgeometry based) correlation and entanglement measures [14,40] for all ξ-correlation and ξ-entanglement. These are the most natural generalizations of the mutual information [6,7], the entanglement entropy [16], and the entanglement of formation [17] for Level II of the multipartite case.
The ξ-correlation of a state is its distinguishability by the relative entropy (1b) from the ξ-uncorrelated states [14,40], With the same reasoning as in section II B, the ξentanglement of a pure state is and for mixed states, one can use the convex roof extension to define the ξ-entanglement of formation We have that C ξ is a correlation monotone (for the proof, see appendix B 1), E ξ is a strong entanglement monotone [40], and both of these are faithful, [40], moreover, they show the same lattice structure as the partition ideals, P II , that is, which is called multipartite monotonicity for Level II [14,40].

D. Level III: classes
The partial correlation and entanglement properties form an inclusion hierarchy (9a)-(9b), for example, if a state is 1|2|3-separable, then it is also 12|3-separable. We are interested in the labeling of the strict, or exclusive properties, for example, those states which are 12|3separable and not 1|2|3-separable. In general, we would like to determine all the possible nonempty intersections of the state sets D ξ-unc and D ξ-sep . We call these intersections partial correlation and partial entanglement classes, containing states of well-defined partial correlation and partial entanglement properties.
To embed these requirements in the labeling of the strict properties of multipartite correlation and entanglement, we use the nonempty up-sets (nonempty filters) of nonempty down-sets (nonempty ideals) of partitions [40], which are sets of down-sets of partitions ξ = {ξ 1 , ξ 2 , . . . , ξ |ξ| } ⊆ P II which are closed upwards with respect to . That is, if ξ ∈ ξ, then for all partition ideals υ ξ we have υ ∈ ξ. The set of the nonempty upsets of nonempty down-sets of partitions of L is denoted with this possesses a lattice structure with respect to the standard inclusion as partial order, which we also call refinement, and also denote with , (For illustrations, see figure 1.) Note that here, contrary to Level I and II, we call ξ finer and υ coarser. In the generic case, if the inclusion of sets is described by a poset P , then the possible intersections can be described by O ↑ (P ) (for the proof, see appendix A.2 in [41]). One may make the classification coarser by selecting a subposet P II* ⊆ P II , with respect to which the classification is done, P III* := O ↑ (P II* ) \ {∅} [40,41].
With respect to the partition ideal filters ξ ∈ P III* , we can define the strict partial correlation and entanglement properties.
The strictly ξ-uncorrelated states are those which are uncorrelated with respect to all ξ ∈ ξ, and correlated with respect to all ξ ∈ ξ = P II* \ ξ, so the class of these (partial correlation class) is The strictly ξ-separable states are those which are separable with respect to all ξ ∈ ξ, and entangled with respect to all ξ ∈ ξ, so the class of these (partial separability class, or partial entanglement class) is The meaning of the Level III hierarchy could also be clarified. If there exists a ∈ C υ-unc and an LO mapping it into C ξ-unc , then υ ξ [40,41], and if there exists a ∈ C υ-sep and an LOCC mapping it into C ξ-sep , then υ ξ [40]. In this sense, the Level III hierarchy compares the strength of correlation and entanglement among the classes.
The filters ξ are sufficient for the description of the nontrivial classes (nonempty intersections) of the correlation and entanglement properties, but they are not necessary in general. For the strictly ξ-uncorrelated states, the structure P III* simplifies significantly [41]. (For example, for the finest classification P II* = P II , we have that the structure of the partial correlation classes is P III* ∼ = P ∂ I , as the unique labeling of the nontrivial partial correlation classes is given as ξ = ↑{↓{ξ}} for the partitions ξ ∈ P I [41].) On the other hand, it is still a conjecture that ξ-separability is nontrivial for all ξ ∈ P III [40]. The conjecture holds for three subsystems, for which explicit examples were constructed [39,69].

III. MULTIPARTITE CORRELATION AND ENTANGLEMENT: PERMUTATION INVARIANT PROPERTIES
Here we build up the structure of the classification and quantification of the permutation invariant multipartite correlation and entanglement properties. We do this from the point of view of the general properties in the previous section.
In section II, for the description of the structure of multipartite correlations [14,40,41], the natural language was that of set partitions [44]. For the permutationinvariant case, particularly k-partitionability and kproducibility, it is natural to use the simpler language of integer partitions [45,58].
A. Level 0: subsystem sizes Let us have the map, simply measuring the sizes of the subsystems X ∈ P 0 , mapping P 0 →P 0 , which is simplŷ This possesses the natural order ≤, which is now denoted with , with respect to which s is monotone, Y X ⇒ s(Y ) s(X). We note for the sake of the analogy with the construction in the subsequent levels that one could define onP 0 as y x if there exist Y ∈ s −1 (y) and X ∈ s −1 (x) such that Y X, from which the monotonicity of s follows automatically (see (A3) in appendix A 1).

B. Level I: integer partitions
Let the map s in (14) act elementwisely on the partitions ξ ∈ P I , where on the right-hand side a multiset stays, that is, a set, which allows multiple instances of its elements. (17a) Its size is given by the partition numbers [70]. Based on the refinement of the set partitions (2b), we define a partial order over the integer partitions, which we also call refinement, and also denote with , given aŝ element ⊥ = {1, 1, . . . , 1} and = {n}. By construction, s is monotone with respect to these partial orders, With respect to the integer partitionsξ ∈P I , we can define the partial correlation and entanglement properties, as well as the measures quantifying them.
Theξ-uncorrelated states are those which are ξuncorrelated (3a) with respect to a partition ξ of typê ξ, which is actually a Level II state set (8a) labeled by the ideal ξ = ↓ s −1 (ξ), the others areξ-correlated states. Theξ-uncorrelated states are exactly those which can be prepared from uncorrelated states by ξ-LO for a partition ξ of typê ξ. (Note that we consider arbitrary states here, not only permutation invariant ones. The correlation property defined in this way is what permutation invariant is.) Theξ-separable states are convex combinations of ξ-uncorrelated states, which is actually a Level II state set (8b) labeled by the ideal ξ = ↓ s −1 (ξ), the others areξ-entangled states. Theξ-separable states are exactly those which can be prepared from uncorrelated states by mixtures of ξ-LOs for different partitions ξ of typeξ. We also have that Dξ -unc is closed under LO, Dξ -sep is closed under LOCC, because these hold for the general Level II state sets D ↓ s −1 (ξ)-unc and D ↓ s −1 (ξ)-sep in section II C. It also follows that these properties show the same partially ordered structure as the integer partitions, (These come by using (A14a) and (A13a) in appendix A 4, then (9) and (18).) In accordance with the general case in section II B, one can define the corresponding (information-geometry based) correlation and entanglement measures for allξcorrelation andξ-entanglement. These are the most natural generalizations of the mutual information [6,7], the entanglement entropy [16], and the entanglement of formation [17] for Level I of the permutation invariant multipartite case.
Theξ-correlation of a state is its distinguishability by the relative entropy (1b) from theξ-uncorrelated states, which is actually a Level II measure (10a) labeled by the ideal ξ = ↓ s −1 (ξ), With the same reasoning as in section II B, theξentanglement of a pure state is and for mixed states, one can use the convex roof extension, which is actually a Level II measure (10c) labeled by the ideal ξ = ↓ s −1 (ξ), These inherit the properties of the general case in section II C. So we have that Cξ is a correlation monotone, Eξ is a strong entanglement monotone, and both of these are faithful, Cξ( ) = 0 ⇔ ∈ Dξ -unc , Eξ( ) = 0 ⇔ ∈ Dξ -sep , moreover, they show the same partially ordered structure as the integer partitions,P I , that is, which we call multipartite monotonicity for Level I of the permutation invariant case. (These come by using (A14a) and (A13a) in appendix A 4, then (11) and (20).)

C. Level II: multiple integer partitions
Let the map s in (16) act elementwisely on the partition ideals ξ ∈ P II , where on the right-hand side a set stays. We call s(ξ) the type of the set partition ideal ξ. This is a nonempty ideal of integer partitions, This might not seem to be obvious, for the proof, see (A12) in appendix A 4. In particular, all ξ ∈ ξ is of typeξ for aξ ∈ s(ξ), on the other hand, allξ ∈ s(ξ) has representative ξ ∈ ξ of typeξ. Based on the refinement of the set partition ideals (7b), we define a partial order over the integer partition ideals, which we also call refinement, and also denote with , given aŝ which turns out to be the inclusion, being the natural partial order for ideals. This might not seem to be obvious, for the proof, see (A12) in appendix A 4. (For illustrations, see figure 4.) Because of these,P II with this partial order is a lattice. By construction, s is monotone with respect to these partial orders, υ ξ ⇒ s(υ) s(ξ) (see (A3) in appendix A 1).
With respect to the integer partition idealsξ ∈P II , we can define the partial correlation and entanglement properties, as well as the measures quantifying these.
Theξ-uncorrelated states are those which are ξuncorrelated (3a) with respect to a partition ξ of typê ξ for aξ ∈ξ, which is actually a Level II state set (8a) labeled by the ideal ξ = ∨s −1 (ξ) = ξ ∈ξ ↓ s −1 (ξ), (for the proof, see (A13b) in appendix A 4); the others arê ξ-correlated states. (We use the notation ∨A := a∈A a for any subset A of a lattice.) Theξ-uncorrelated states are exactly those which can be prepared from uncorrelated states by ξ-LO for a partition ξ of a typeξ ∈ξ.
Theξ-separable states are convex combinations ofξuncorrelated states, which is actually a Level II state set (8b) labeled by the ideal ξ = ∨s −1 (ξ) = ξ ∈ξ ↓ s −1 (ξ), the others areξ-entangled states. Theξ-separable states are exactly those which can be prepared from uncorrelated states by mixtures of ξ-LOs for different partitions ξ of different typesξ ∈ξ. We also have that Dξ -unc is closed under LO, Dξ -sep is closed under LOCC, because these hold for the general Level II state sets D ∨s −1 (ξ) and D ∨s −1 (ξ) in section II C. It also follows that these properties show the same lattice structure as the integer partition ideals,P II , that is, (These come by using (A14b) and (A13b) in appendix A 4, then (9) and (24).) In accordance with the general case in section II C, one can define the corresponding (information-geometry based) correlation and entanglement measures for allξcorrelation andξ-entanglement. These are the most natural generalizations of the mutual information [6,7], the entanglement entropy [16], and the entanglement of formation [17] for Level II of the permutation invariant multipartite case.
Theξ-correlation of a state is its distinguishability by the relative entropy (1b) from theξ-uncorrelated states, which is actually a Level II measure (10a) labeled by the ideal ξ = ∨s −1 (ξ), With the same reasoning as in section II C, theξentanglement of a pure state is and for mixed states, one can use the convex roof extension, which is actually a Level II measure (10c) labeled by the ideal ξ = ∨s −1 (ξ), These inherit the properties of of the general case in section II C. So we have that Cξ is a correlation monotone, Eξ is a strong entanglement monotone, and both of these are faithful, Cξ( ) = 0 ⇔ ∈ Dξ -unc , Eξ( ) = 0 ⇔ ∈ Dξ -sep , moreover, they show the same lattice structure as the integer partition ideals,P II , that is, which we call multipartite monotonicity for Level II of the permutation invariant case. (These come by using (A14b) and (A13b) in appendix A 4, then (11) and (26).)

D. Level III: classes
Let the map s in (22) act elementwisely on the partition ideal filters ξ ∈ P III , where on the right-hand side a set stays. We call s(ξ) the type of the set partition ideal filter ξ. This is a nonempty filter of ideals of integer partitions, This might not seem to be obvious, for the proof, see (A12) in appendix A 4. Based on the refinement of the set partition ideal filters (12b), we define a partial order over the integer partition ideal filters, which we also call refinement, and also denote with , given aŝ which turns out to be the inclusion, being the natural partial order for filters. This might not seem to be obvious, for the proof, see (A12) in appendix A 4. (For illustrations, see figure 4.) Note that here, contrary to Level I and II, we callξ finer andυ coarser. Because of these,P III with this partial order is a lattice. By construction, s is monotone with respect to these partial orders, υ ξ ⇒ s(υ) s(ξ) (see (A3) in appendix A 1).
With respect to the integer partition ideal filtersξ ∈ P III , we can define the strict partial correlation and entanglement properties.
The strictlyξ-uncorrelated states are those which are uncorrelated with respect to allξ ∈ξ, and correlated with respect to allξ ∈ξ =P II \ξ, so the class of these (permutation invariant partial correlation class) is The strictlyξ-separable states are those which are separable with respect to allξ ∈ξ, and entangled with respect to allξ ∈ξ, so the class of these (permutation invariant partial separability class, or permutation invariant partial entanglement class) is The meaning of the permutation invariant Level III hierarchy can also be clarified. If there exists a ∈ Cυ

IV. AN ALTERNATIVE WAY OF INTRODUCING PERMUTATION INVARIANCE
The Level II structure P II encodes the different partial correlation and entanglement properties. In sections II and III, we have built up the structure of the classification, parallel for the general, and for the permutation invariant cases. Here we show a more compact, but less transparent treatment of the same structure, by embedding both constructions into P II .
The three-level building of the correlation and entanglement classification was given by the construction in section II. Because of (4), one may embed P I into P II , using the principal ideals in P II , P I ∼ = P II pr. := ↓{ξ} ξ ∈ P I ⊆ P II , by noting that υ ξ ⇐⇒ ↓{υ} ↓{ξ}. In this way, P I is a sublattice of P II . The same can be done for the permutation invariant correlation and entanglement classification given by the construction in section III. In the same way, because of (19), one may embedP I intoP II , using the principal ideals inP II , by noting thatυ ξ ⇐⇒ ↓{υ} ↓{ξ}.
In this way,P I is a sublattice ofP II . The third point here is that, noting that ↓ s −1 (ξ) = ∨s −1 (↓{ξ}) (for the proof, see (A13a) in appendix A 4), one may also embedP II (thus alsoP I ) into P II , usinĝ by noting that (For the proof, see (A14a) and (A14b) in appendix A 4.) In this way,P I andP II are sublattices of P II . A ξ ∈ P II is permutation invariant, if and only if ∨s −1 (s(ξ)) = ξ. (Then ξ ∈ P II describes the same property asξ := s(ξ) ∈P II .) So P II pinv. can be given directly as and the permutation invariant classification can be described as a coarsened classificationP III ∼ = P III* = O ↑ (P II* ) \ {∅} with respect to the permutation invariant properties P II* = P II pinv. . We note that P II could also be embedded into P III by principal filters (and similarlyP II intoP III ), however, the construction has led to a different direction.

V. k-PARTITIONABILITY, k-PRODUCIBILITY AND k-STRETCHABILITY
In this section, we consider k-partitionability and kproducibility as particular cases of permutation invariant properties of partial correlation and entanglement, and elaborate a duality between them. Our point of view makes possible to reveal a new property, which we call k-stretchability, combining some advantages of kpartitionability and k-producibility. We also investigate the relations among these three properties.   A. k-partitionability and k-producibility of correlation and entanglement k-partitionability and k-producibility are permutation invariant Level II properties. A partition is kpartitionable, if the number of its parts is at least k, while it is k-producible, if the sizes of its parts are at most k. (For illustration, see figure 5.) These are encoded by the ideals of k-partitionable and k-producible partitions, ξ k-part := ξ ∈ P I |ξ| ≥ k ∈ P II , (35a) ξ k-prod := ξ ∈ P I ∀X ∈ ξ : |X| ≤ k ∈ P II , (35b) for k = 1, 2, . . . , n, forming chains in the lattice P II , ξ l-prod ξ k-prod ⇐⇒ l ≤ k.
Since k-partitionability and k-producibility are permutation invariant properties, it is enough to consider their types (22) ξ k-part := s(ξ k-part ) = ξ ∈P I |ξ| ≥ k ∈P II , (37a) forming chains in the latticeP II , The corresponding k-partitionably uncorrelated and kproducibly uncorrelated states (24a) are D k-part unc := D ξ k-part -unc = Dξ k-part -unc = X∈ξ X ∀ξ ∈ P I s.t. |ξ| ≥ k , (with X ∈ D X ), which are products of density operators of at least k subsystems, and of density operators of subsystems containing at most k elementary subsystems, respectively. The corresponding k-partitionably separable (also called k-separable [35,37,47]) and k-producibly separable (also called k-producible [46,47,49]) states (24b) are which can be decomposed into k-partitionably, or kproducibly uncorrelated states, respectively. Because of (25a), these properties show the same chain structure as the chains of the corresponding partition ideals (38), that is, that is, if a state is l-partitionably uncorrelated (separable) then it is also k-partitionably uncorrelated (separable) for all l ≥ k, and if a state is l-producibly uncorre-lated (separable) then it is also k-producibly uncorrelated (separable) for all l ≤ k. The corresponding k-partitionability correlation and kproducibility correlation (26a) are [14] C k-part := C ξ k-part = Cξ The corresponding k-partitionability entanglement and k-producibility entanglement (26c) are [40] E k-part := E ξ k-part = Eξ Because of (27), these measures show the same chain structure as the chains of the corresponding partition ideals (38), that is, that is, l-partitionability correlation (entanglement) is always stronger than k-partitionability correlation (entanglement) for all l ≥ k, and l-producibility correlation (entanglement) is always stronger than k-producibility correlation (entanglement) for all l ≤ k, which is the multipartite monotonicity of these measures.

B. Duality by conjugation
Looking at the k-partitionability and k-producibility properties in figure 5, and also at their definitions (35a) and (35b), it is not clear how these are related to each other.
In section III, the permutation invariant correlation properties were described by the use of integer partitions, represented by Young diagrams in the figures. Important properties of Young diagrams are their height, width and rank, given for an integer partitionξ ∈P I as and also for set partition ξ ∈ P I as h(ξ) := h(s(ξ)) = |ξ|, w(ξ) := w(s(ξ)) = max X∈ξ |X|, r(ξ) := r(s(ξ)) = w(ξ) − h(ξ). It is enlightening now to arrange the poset P I according to the height and width of the partitions, in the way as can be seen in figure 6, i.e., the height is increasing downwards, the width is increasing to the right, then the rank is increasing up-right. It is easy to see that the height is strictly decreasing, the width is increasing, and the rank is strictly increasing monotone with respect to the refinement (17b), (For illustration, see figure 6: the arrows point always upwards, and possibly to the up-right, never to the left or downwards.) Now the partitionability and producibility properties (35a) and (35b) can be formulated with the height and width asξ so k-partitionability is the minimal height, kproducibility is the maximal width of the partition. In the height-width based arrangement ofP I in figure 6, the k-partitionable partitions (45a) are below the k-th rows (counting from the top), and k-producible partitions (45b) are to the left from the k-th column (counting from the left). We can also introduce another property, called k-stretchability, aŝ for k = −(n − 1), −(n − 2), . . . , n − 2, n − 1, so kstretchability is the maximal rank of the partition. It is easy to see thatξ k-str ∈P II , that is, it is a down-set for all k, by the third inequality in (44). We elaborate on the arising correlation and entanglement properties later, in section V C. On the set of Young diagrams, an involution arises naturally, called conjugation, being the reflection of the diagram with respect to its "diagonal" [45]. Considering the integer partitions themselves, the conjugation is given as follows. For decreasingly ordered values (x 1 , x 2 , . . . , x |ξ| ) of the elements ofξ, the number of elements x that equal to i is x i −x i+1 in the conjugated partition (setting x i+1 = 0 for i = h(ξ)). With multisets, the conjugation is given as the mapP I →P I , (46) where both sets on the right-hand side are multisets.
The partial order (17b) does not show nice properties with respect to the conjugation.  ordered by refinement (the first example is for n = 6, where we have {2, 2, 2} and {3, 3}, which are conjugates of each other, and cannot be ordered).
The conjugation interchanges the height and width, and multiplies the rank with −1, (47) (For illustration, see the height-width based arrangement in figure 6: the conjugation brings to the position mirrored with respect to the diagonal w(ξ) = h(ξ).) Note that, although the conjugation interchanges the height and width, it does not interchange the partitionability (45a) and producibility (45b) properties, since these are given in different ways by height and width, namely, by lower and upper bounds.

C. k-stretchability of correlation and entanglement
In section V B, we introduced the permutation invariant property k-stretchability (45c) for k = −(n − 1), −(n − 2), . . . , n − 2, n − 1. Because of the third inequality in (44), these form a chain in the latticeP II , (For illustration, see the height-width based arrangement in figure 6.) The corresponding k-stretchably uncorrelated states (24a) are which can be decomposed into k-stretchably uncorrelated states. Because of (25a), these properties show the same chain structure as the chains of the corresponding partition ideals (48), that is, that is, if a state is l-stretchably uncorrelated (separable) then it is also k-stretchably uncorrelated (separable) for all l ≤ k.
The corresponding k-stretchability correlation (26a) is The corresponding k-stretchability entanglement (26c) is Because of (27), these measures show the same chain structure as the chain of the corresponding partition ideals (48), that is, that is, l-stretchability correlation (entanglement) is always stronger than k-stretchability correlation (entanglement) for all l ≤ k, which is the multipartite monotonicity of these measures. k-partitionability (45a), k-producibility (45b) and kstretchability (45c) give, of course, a rather coarsened description of permutation invariant correlation or entanglement properties. Because of this, they must show some disadvantages. First, k-partitionability (45a) takes into account only the number of subsystems uncorrelated (or separable) from one another, and does not distinguish between the cases when subsystems of roughly equal sizes are uncorrelated (or separable), and when some elementary subsystems are uncorrelated (or separable) from the rest of a large system, although these two cases represent highly different situations form a resource-theoretical point of view. (For illustration, see the rows in figure  6.) Second, k-producibility (45b) takes into account only the size of the largest subsystem uncorrelated (or separable) from the other part of the system, and does not distinguish between the cases when a large subsystem is uncorrelated (or separable) from a slightly smaller subsystem, or from many elementary subsystems, although these two cases represent again highly different situations form a resource-theoretical point of view. (For illustration, see the columns in figure 6.) Third, k-stretchability (45c) combines the advantages of the previous two properties in a balanced way, it takes into account a kind of "difference" of them, and does not distinguish among cases with rather different types. (For illustration, see the lines parallel to the diagonal in figure 6.) D. Relations among k-partitionability, k-producibility and k-stretchability The height h(ξ), width w(ξ) and rank r(ξ) (43) of the integer partitionsξ ∈P I of n ∈ N are bounded by one another as The first inequality in (53a) can be seen by noting that n ≤ hw, because the right-hand side is the area of the smallest rectangle into which the Young diagram of n boxes fits. The second inequality in (53a) can be seen by noting that h + w ≤ n + 1, because the left-hand side is the half circumference of the smallest rectangle into which the Young diagram of n boxes fits. The inequalities in (53b) come by the same reasoning, or by conjugation (47) in (53a). The inequalities in (53c) come by subtracting h from (53b). The inequalities in (53d) come by an analogous reasoning, or by conjugation (47) in (53c). The inequalities in (53e) come by solving the respective inequalities in (53c) for h. The inequalities in (53f) come by an analogous reasoning, or by conjugation (47) in (53e). The first inequality in (53d) and the second inequalities in the others in (53) are saturated for the partition ξ = {m, 1, 1, . . . , 1}, where the integer 1 occurs n − m times. The second inequality in (53d) and the first inequalities in the others in (53) are saturated for the partitionξ = {m, m, . . . , m, n − ( n/m − 1)m}, where the integer m occurs n/m − 1 times. In the cases where noninteger value stays on one side of an inequality, saturation is understood for the inequality strengthened by the ceiling or floor functions x = min{m ∈ Z | m ≥ x} or x = max{m ∈ Z | m ≤ x}; that is, if q ≤ m ∈ N, then also q ≤ m, and if q ≥ m ∈ N, then also q ≥ m. (For illustration, see figure 6. The second inequalities in (53a) and (53b) express that the arrangement of Young diagrams in figure 6 is "skew upper triangular", while the first inequalities in (53a) and (53b) describe the "hyperbolic" shape of the upper boundary. The other in-equalities in (53) give the boundaries in the rank-height or width-rank plane.) Since the partitionability is related to lower-bounding the height (45a), the producibility is related to upperbounding the width (45b), and the stretchability is related to their difference (45c), the bounds in (53) lead to the following relations between these propertieŝ The relations in (54a) and (54b) can be seen by using the second inequality in (53b) and the second inequality in (53c), respectively, with (45a). The relations in (54c) and (54d) can be seen by using the first inequality in (53a) and the second inequality in (53d), respectively, with (45b). (For noninteger values, strengthening by the ceiling function was also exploited, as before.) The relations in (54e) and (54f) can be seen by using the first inequality in (53e) and the second inequality in (53f), respectively, with (45c). From the relations (54), for the inclusion of the state spaces (39), we have D k-prod unc ⊆ D n/k -part unc , (55c) D k-str unc ⊆ D 1 2 (n+1+k)-prod unc , (55f) by the order isomorphisms (25); and for the bounds of the measures (41), we have by the multipartite monotonicity (27).

VI. SUMMARY, REMARKS AND OPEN QUESTIONS
In this work we investigated the partial correlation and entanglement properties which are invariant under the permutations of the subsystems. The set partition based three-level structure, describing the classification of partial correlation and entanglement (section II) was mapped to a parallel, integer partition based three-level structure, describing the permutation invariant case (section III). This mapping is easy to understand on Level I of the construction, however, to see that it is working well through the whole construction, a formal proof was given (appendix A). The construction can be made more compact, although less transparent (section IV), which can be used as a starting point of more advanced investigations.
We also investigated k-partitionability and kproducibility, fitting naturally into the structure of permutation invariant properties. A kind of combination of these two gives k-stretchability, which is sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated (or separable) from one another. We studied their relations, and a duality, connecting the former two.
In the following, we list some remarks and open questions.
The first point to note is that we have followed a treatment of entanglement, which is somewhat different than the standard LOCC paradigm [15]. The LOCC paradigm grabs the essence of entanglement as a correlation which cannot be created or increased by classical communication (classical interaction). This point of view leads to a classification too detailed and practically unaccomplishable for the multipartite scenario. Our treatment is rooted more in statistics, by noticing that (i) pure states of classical systems are always uncorrelated, so in pure states, correlations are of quantum origin, and this is what we call entanglement [1]; and (ii) mixed states of classical systems can always be formed by forgetting about the identity of pure (hence uncorrelated) states, so in mixed states, correlations which cannot be described in this way are of quantum origin, and this is what we call entanglement. These principles are working painlessly in the multipartite scenario, entangled states and correlation based definitions of entanglement measures were defined based on these in the first two levels of the structure of multipartite entanglement. In particular, from (i) it follows that entanglement in pure states should be measured by correlation (see (5b) and (10b), and the corresponding quantities for the permutation invariant case, and the notes in the end of section II B). We emphasize that this statistical point of view is fully compatible with LOCC, and has led to a much coarser, finite classification.
Being uncertain (or forgetting) about the identity of the state of the system is a guiding principle in the definition of the different aspects of multipartite entanglement in section II (see also in point (ii) in section VII.A. in [40]). States in D L : We are uncertain about the (pure) state, by which the system is described (section II A). States in D ξ-unc (D ξ-sep ): We are uncertain about the pure state, by which the system is described, but we are certain about the partition with respect to which the state is uncorrelated (separable) (section II B). States in D ξ-unc (D ξ-sep ): We are uncertain about the pure state, by which the system is described, and we are also uncertain about the partition with respect to which the state is uncorrelated (separable), but we are certain about the possible partitions with respect to which the state is uncorrelated (separable) (section II C). States in C ξ-unc (C ξ-sep ): We are uncertain about the pure state, by which the system is described, and we are also uncertain about the partition with respect to which the state is uncorrelated (separable), but we are certain about the possible partitions with respect to which the state is uncorrelated (separable), and we are also certain about the possible partitions with respect to which the state is correlated (entangled) (section II D).
We also mention here that for the state sets of given multipartite correlation (entanglement) properties D ξ-unc (D ξ-sep ) we cannot formulate semigroups of quantum channels which could play the role of "free operations" for these "free state" sets in a usual resource-theoretical scenario [71]. Instead of this, being uncertain (or forgetting) about the identity of the maps is the guiding principle here (see the characterizations in sections II B and II C, as well as in the permutation invariant case in sections III B and III C). Note that this is physical: only the fully uncorrelated (fully separable) states are for free, the others are resource states, possibly of different "strength." Note that, after the general case (section II), we considered the permutation invariant (correlation or entanglement) properties (from section III). The use of these is not restricted to permutation invariant states: these are simply the permutation invariant properties of also non-permutation-invariant states. If the states considered are permutation invariant (for example, in the first quantization of bosonic or fermionic systems), then these are the only relevant properties.
The permutation invariant properties were described by the use of integer partitions, for which a refinementlike order was given, which is just the coarsening of the refinement order of set partitions. In the literature, there are several partial orders constructed for integer partitions, such as the dominance order [72] (based on majorization, leading to a lattice), the Young order [45,58] (based on diagram containment, given for partitions of all integers, leading to Young's lattice, here partitions of the same integer cannot be ordered, and the order is invariant to the conjugation), or the reverse lexicographic order (leading to a chain). The refinement order for integer partitions, introduced in (17b) through the refinement order for set partitions (2b), is different from the above orders, and, to our knowledge [45,58], was not considered upon its merits in the literature before. Although Birkhoff mentioned this partial order in his book [73] as an example, it was not used for any reasonable purpose. The reason for this might be that the resulting poset shows properties not so nice or powerful as those shown by the others. However, this order is what needed is in the classification problem of permutation invariant properties, what is reflected also by the monotonicities (44).
k-partitionability is about the natural gradation of the lattice of set partitions P I or of the poset of integer par-titionsP I , but it was not clear, how k-producibility can be understood. The conjugation of integer partitions has explained the role of the latter, by establishing a kind of duality between the two properties. Note, however, that this duality is only partial: although the conjugation of integer partitions (46) interchanges the height and the width (47), by which the k-partitionability and kproducibility properties are given, but the latter properties are given in an opposite way by height and width, see (45a) and (45b). So the conjugation does not interchange k-partitionability and k-producibility, it establishes a connection on the deeper level of integer partitions only. This is also reflected in the relations among the different k-partitionability and k-producibility properties, given in (54a) and (54c).
For n ≤ 6, the height and width determineξ uniquely. For n = 7, we have {3, 2, 2} and {3, 3, 1}, being of height 3 and width 3. Note that two different integer partitions of the same height and width can never be ordered. (Ordered pairs are of different height, because of the strict monotonicity of the height (44).) The property of k-stretchability (45c) was introduced by the rank of the partition. We note that the rank of an integer partition (43c) was defined and used originally in a very different context in number theory and combinatorics. It was introduced by Freeman Dyson [74] in his investigations of Ramanujan's congruences in the partition function [70].
For n subsystems, the number of nontrivial kpartitionability and k-producibility properties is n − 1 in both cases (1-partitionability and n-producibility are trivial), while the number of nontrivial k-stretchability properties is 2(n − 1) ((n − 1)-stretchability is trivial). All of these three properties form chains, see (38a), (38b) and (48), and the relations among them are given in (54).
Note that k-stretchability combines the advantages of kpartitionability and k-producibility, in the sense that it rewards large correlated (or entangled) subsystems, while it punishes the larger number of subsystems uncorrelated (or separable) with one another. Also, while the relations between k-partitionability and k-producibility are highly nontrivial (54a), (54c), the k-stretchability properties form a chain (48). The price to pay for this is that sometimes there are rather different properties not distinguished by stretchability, for example, {3, 3} and {4, 1, 1} are both 1-stretchable. Taking stretchability seriously leads to a highly nontrivial, balanced comparison of permutation invariant multipartite correlation or en-tanglement properties. that b = f (b) = f (a) = a . (This proof uses (A2a) and (A2c). A similar proof can be given by using (A2b) and (A2c).) (iii) Transitivity (∀a , b , c ∈ P , if c b and b a , then c a ): let b a , which means that there exist a ∈ f −1 (a ) and b ∈ f −1 (b ) such that b a, by (A1). Fix such a pair b and a. Also, let c b , applying (A2a) for b, we have that there exists c ∈ f −1 (c ), such that c b. Then, since the partial order is transitive, we have that c a for an a ∈ f −1 (a ) and c ∈ f −1 (c ), which means that c a by (A1). (This proof uses (A2a). A similar proof can be given by using (A2b).) The following construction would be much simpler if the conditions (A2) would also be necessary. Note that this is not the case: if is a partial order, then (A2c) holds, but (A2a) and (A2b) do not hold. For example, for the poset P = {a 1 , a 2 , b} with the only arrow b ≺ a 1 , and the function given as f (a 1 ) = f (a 2 ) = a = f (b) = b , we have b ≺ a , but (A2a) does not hold.
If (A2a) or (A2b), and (A2c) hold, then the new poset (P , ) can be considered as a coarsening of (P, ) by f . (f : P → P is surjective by definition. If it is also injective, then (P, ) and (P , ) are isomorphic, and the coarsening is trivial.) Note that, because of the construction, f : P → P is automatically monotone, holds.) Note also that, if the poset (P, ) is a lattice, the transformed one (P , ) is not necessarily that. To make it a lattice, further conditions have to be imposed; this problem is not addressed here.

Down-sets and up-sets
In a poset P , a down-set (order ideal ) is a subset a ⊆ P , which is closed downwards [43,45], ⇐⇒ ∀a ∈ a, ∀b a, b ∈ a; (A4a) while an up-set (order filter ) is a subset a ⊆ P , which is closed upwards [43,45], These form lattices with respect to the inclusion ⊆, with the join ∨ (least upper bound) being the union ∪ and the meet ∧ (greatest lower bound) being the intersection ∩. (A5b) (The existence of the bottom elements in P and P ensures that not only the down-sets but also the nonempty down-sets form lattices in both cases [40].) Then, by denoting the elementwise action of f with g, that is, g(a) = {f (a) | a ∈ a}, in the following lemmas we will show that (Q , ) = (g(Q), g( )), that is, the diagram (Q, ) g / / (Q , ) commutes. Here g( ) is the partial order transformed by g in the same way as was transformed by f in (A1), and we also have automatically that g is monotone for and g( ), in the same way as in (A3).
The following two technical lemmas concerning the sets a ∈a f −1 (a ) ⊆ P will be used several times later. Lemma 2. In the above setting (and assuming (A2)), for all a ∈ Q , we have g a ∈a f −1 (a ) = a .
Proof. This is because g is the elementwise action of f , so g a ∈a f −1 (a ) = g a ∈a {a ∈ P | f (a) = a } = a ∈a g {a ∈ P | f (a) = a } = a ∈a {a } = a . Lemma 3. In the above setting (and assuming (A2)), for all a ∈ Q , we have a ∈a f −1 (a ) ∈ Q, that is, it is a nonempty down-set.
Proof. First, denote a := a ∈a f −1 (a ). On the one hand, a = ∅, since a = ∅ and f is surjective. On the other hand, if b a for an a ∈ a, then f (b) f (a) because of the monotonicity (A3). Since f (a) ∈ a by the construction of a, and a is a down-set (A4a), we have that f (b) ∈ a . Then b ∈ f −1 (f (b)) ⊆ a by the construction of a = f −1 (f (b)) ∪ a ∈a ,a =f (b) f −1 (a ), and then b ∈ a, so a is a down-set. Altogether we have that a ∈ Q.
Proof. We need to prove both inclusions. (i) We need that g(Q) ⊆ Q . Let a ∈ Q and a := g(a). Then for all a ∈ a let us have a ∈ f −1 (a ) ∩ a. (Note that f −1 (a ) ∩ a = ∅ by construction.) Then let b a , and applying (A2a) for a, we have that there exists b ∈ f −1 (b ) such that b a. Since a ∈ a, and a is a down-set (A4a), we have that b ∈ a, and then b ∈ a , so a is a down-set. (ii) We also need that g(Q) ⊇ Q . Let a ∈ Q , then we construct an a ∈ Q, for which a = g(a). The element a := a ∈a f −1 (a ) fulfills these criteria, we have a = g(a) by Lemma 2, and a ∈ Q by Lemma 3.
Proof. We need to prove both directions. (i) We need that for all a , b ∈ Q , if b g( )a then b a . b g( )a means that ∃b ∈ g −1 (b ) and ∃a ∈ g −1 (a ) such that b a. (Again, g −1 (a ) = {a ∈ Q | g(a) = a } is the inverse image of the singleton {a }.) In this case, b = g(b) = {f (b) | b ∈ b} ⊆ {f (a) | a ∈ a} = g(a) = a , so b a . (ii) We also need that for all a , b ∈ Q , if b a then b g( )a . From b and a , let us form b := b ∈b f −1 (b ) and a := a ∈a f −1 (a ). We have that b ∈ Q and a ∈ Q (they are nonempty down-sets) by Lemma 3, also g(b) = b and g(a) = a by Lemma 2, and if b a then b a by construction. That is, we have constructed b ∈ g −1 (b ) and a ∈ g −1 (a ), for which b a, so we have b g( )a .
Recall also that if we form the down-set lattices (A5), then Lemma 4 and Lemma 5 together state that (A6) commutes, Lemma 6 states that the conditions in (A2) are inherited. Then with Lemma 7, we have that in the main text, the three-level construction is well-defined as follows.
Corollary 17. In the setting of the main text, the following diagram commutes (P III , ) s / / (P III , ) With Lemma 12 and (A10), we also have that for the first two levels of the construction in the main text, the following identities concerning the principal ideals hold.