On the coset category of a skew lattice

Skew lattices are non-commutative generalizations of lattices. The coset structure decomposition is an original approach to the study of these algebras describing the relation between its rectangular classes. In this paper we will look at the category determined by these rectangular algebras and the morphisms between them, showing that not all skew lattices can determine such a category. Furthermore, we will present a class of examples of skew lattices in rings that are not strictly categorical, and present sufficient conditions for skew lattices of matrices in rings to constitute $\wedge$-distributive skew lattices.


Introduction
Skew lattices are one of the most successful generalizations of lattices, being non commutative but maintaining associativity, idempotency and four of the several possible absorption laws. The chosen absorption laws permit us to generalize several lattice theoretic concepts as is the case of distributivity, studied in [11] or [16]. The order structure of skew lattices finds a close relation to the order structure of its corresponding lattice. These algebras can also be seen as double bands due to the fact that their reducts (S; ∧) and (S; ∨) are regular semigroups of idempotents. Green's relations take an important role in this research. In particular, D is a congruence determining a decomposition deriving from Clifford-MacLean's result, that permits us to look at a skew lattice as a lattice of maximal rectangular algebras.
The study of the coset structure of a skew lattice explores the interplay between related D-classes. It is an approach that has no counterpart in Semigroup Theory or in Lattice Theory. In the lattice case, the D-classes reduce to singletons, while in the case of bands those classes have a known impact in the study of the corresponding semi-lattice. This makes the approach a relevant method to study skew lattices, specifically. Under certain conditions, such an algebra permits the construction of a category that has the D-classes as objects and the coset bijections between them as morphisms. This paper explores questions posed in [18] regarding such a category, named coset category. In this paper we show that all cosets are rectangular subalgebras and that all coset bijections between them are in fact isomorphisms. Those isomorphisms describe the order structure of a skew lattice, as seen in [18]. This study of the characterization of several subvarieties of skew lattices by identities involving cosets, named coset laws, started in [13] and was continued in [7] and in [5] .
When considering a ring R, the operations defined by x ∧ y = xy and x ∨ y = x + y − xy succeeded in providing a rather large class of examples of skew lattices which have motivated many of the properties studied in the general case. When E(R) is the set of all idempotent elements in a ring R and S ⊆ E(R) is closed under both ∨ and ∧, (S; ∧, ∨) is a skew lattice. Skew lattices in rings, together with skew Boolean algebras, constitute the largest classes of studied examples of skew lattices to date. Much has been done also in the particular case of skew lattices in rings of matrices by Cvetko-Vah and Leech in [7], [3] and [2] following the work of Radjavi and other authors on bands of matrices (see [8] and [9]). It was shown in [7] that skew lattices in rings are not normal. In the last section of this paper we will look at the coset category of skew lattices of matrices in rings and show that, in general, skew lattices in rings are also not strictly categorical. We will also present sufficient conditions for skew lattices of matrices in rings to constitute strictly categorical skew lattices and ∧-distributive skew lattices. This paper will be using: basic knowledge of Lattice Theory that can be retrieved in [1] dealing with lattice notions in a noncommutative context; several issues and results deriving from Semigroup Theory, that can be found in [10], having in mind that we are dealing with bands of semigroups; and some Category Theory language, that can be revisited in [17].

Preliminaries
A skew lattice is a set S with binary operations ∧ and ∨ that are both idempotent and associative, satisfying the absorption laws x∧(x∨y) = x = (y∨x)∧x and their duals. A band is a semigroup of idempotents. Recall that a band is regular if it satisfies xyxzx = xyzx, is normal if it satisfies xyzw = xzyw, and is rectangular if it satisfies xyx = x. Any skew lattice S can be seen as double regular band by considering the band reducts (S, ∧) and (S, ∨). If these bands are rectangular we say that the skew lattice S is rectangular. On the other hand, normal skew lattices are the ones for which (S; ∧) is a normal band, and conormal skew lattices are the ones for which (S; ∨) is a normal band. A skew lattice is symmetric whenever Green's relations are five equivalence relations, introduced in [6], characterizing the elements of a semigroup in terms of the principal ideals they generate. Due to the absorption dualities, the Green's relations in the context of skew lattices are defined in [11] by R = R ∧ = L ∨ , L = L ∧ = R ∨ and D = D ∧ = D ∨ . Right-handed skew lattices are the skew lattices for which R = D while left-handed skew lattices are determined by L = D.
Two distinct concepts of order can be considered in a skew lattice S: the natural partial order defined by x ≥ y if x∧ y = y = y ∧ x or, dually, x∨ y = x = y ∨ x; the natural preorder defined by x y if y ∧ x ∧ y = y or, dually, x ∨ y ∨ x = x. Observe that x D y iff x y and y x. Usually D is referred in the available literature as the natural equivalence.
The fact that D can be expressed by the natural preorder allows us to draw diagrams, based on the Caley tables of the corresponding operations, that are capable of representing skew lattices as the one in Figure 1. An admissible Hasse diagram of (a subset of) a skew lattice is a Hasse diagram for the natural partial order (usually represented by full edges) together with an indication of all D-congruent elements (usually represented by dashed edges). Unlike lattices, one such diagram can represent two distinct skew lattices (cf. [11]). Whenever S is a skew lattice, D is a congruence, S/D is the maximal lattice image of S and all congruence classes of D are maximal rectangular skew lattices in S (cf. [11]). Thus, the functor S → S/D is a reflection of skew lattices into ordinary lattices. Hence, to the lattice image S/D we now call lattice reflection.

On the coset structure
In the following section we shall discuss some aspects of the coset structure of a skew lattice, introduced in [13], and further developed in [18] and [20]. Recall that a chain (or totally ordered set) is a set where each two elements are (order) related, and an antichain is a set where no two elements are (order) related. We call S a skew chain whenever S/D is a chain. All D-classes are antichains for the partial order and chains for the preorder (cf. [11]).
Consider a skew lattice S consisting of exactly two D-classes A > B.
Theorem 1. [13] Let S be a skew lattice with comparable D-classes A > B. Then, B is partitioned by the cosets of A in B and the image set of any element a ∈ A in B is a transversal of the cosets of A in B; dual remarks hold for any b ∈ B and the cosets of Furthermore, the operations ∧ and ∨ on A ∪ B are determined jointly by the coset bijections and the rectangular structure of each D-class.

Proposition 2.
[5] Let S be a skew lattice with comparable D-classes A > B and let y, y ′ ∈ B. The following are equivalent: Dual results hold, having a similar statement.
Lemma 3. Let S be a skew lattice. Then, the rectangularity of ∧ (dually, of ∨) implies the rectangularity of S. Moreover, it is equivalent to the validity of the identity x ∧ y = y ∨ x.
Proof. Let S be a skew lattice and x, y, z ∈ S. Assuming the rectangularity of ∧ (i.e.
where the first equality issue to the assumption and the second to absorption, and also where again the first equality is due to absorption, the second is due to the assumption, and the third follows by idempotency. Thus, we get that The proof that the rectangularity of ∨ implies the rectangularity of the skew lattice is now similar.
Given two partially ordered sets (S, ≤) and (T, ≤), a function f : S −→ T is an orderembedding if f is both order-preserving and order-reflecting, i.e. for all x, y ∈ S, x ≤ y if and only if f (x) ≤ f (y). As for lattices, an order isomorphism can be characterized as a surjective order-embedding. Any order-embedding f restricts to an isomorphism between its domain S and its range f (S). Proof. Let x, y ∈ A and assume that φ is a ∧-homomorphism. Then, due to the rectangularity of A and B, respectively. The converse is analogous.

Proof. Consider two comparable D-classes
if a in B are also closed under ∨ follows from the rectangular identity, x ∧ y = y ∧ x given in Lemma 3. We shall now show that each A-coset in B is isomorphic to any B-coset in A. Let a ∈ A and b ∈ B and consider the bijection φ a,b between B ∨ a ∨ B and due to the regularity of ∧. Hence, φ a,b is an isomorphism due to Lemma 4.
This relation is the equivalence corresponding to the coset partition of B in C. Furthermore, it is a congruence of B. Dually, the equivalence θ [B:C] derived from the coset partition of C in B is a congruence of C.
Proof. Let x, y, z, w ∈ C such that xθ [B:C] y and zθ [B:C] w. Fix b ∈ B. Then, Propo- due to the assumption and to regularity. On the The dual statement has a similar proof.
A categorical skew lattice is strictly categorical if the compositions of coset bijections between comparable D-classes A > B > C are never empty. Rectangular and normal skew lattices are strictly categorical skew lattices (cf. [13]). In particular, subskew lattices of strictly categorical skew lattices are also strictly categorical.

Proposition 9.
[7] A skew chain S consisting of D-classes A > B > C is categorical iff for all elements a ∈ A, b ∈ B and c ∈ C satisfying a > b > c, one (and hence both) of the following equivalent statements holds: Moreover, S is strictly categorical iff in addition to (i)-(ii), for all b, b ′ ∈ B, The following is a practical criteria to identify strictly categorical skew lattices.

Proposition 10. [15] A skew chain A > B > C is strictly categorical if and only if given
Example 11. A minimal example of a categorical skew lattice that is not strictly categorical is given by the right-handed manifestation of the skew chain with three D-classes in Figure  1. In fact, the composition of the coset bijections ψ :

Proposition 12. [19] Let S be a skew lattice. Then, S is normal iff for each comparable pair of D-classes
Dually, S is conormal iff for all comparable pairs of D-classes A > B in S and all x, Example 13. Strictly categorical skew lattices need not be normal: any skew lattice with the admissible Hasse diagram below represents a right-handed skew chain and thus a strictly categorical skew lattice (consider, for instance, the subskew lattice { 1, 2, 3 } of the skew lattice in Example 1.

3
1 Normality fails as the upper D-class determines more then one coset in the lower D-class: observe that, considering A = { 1 } and B = { 2, 3 }, A > B is a strictly categorical skew chain, according to Proposition 10, but The prefix categorical was motivated by the definition of categorical skew lattices as the ones for whom coset bijections form a category under certain conditions: being strictly categorical. This category was first introduced in [13], mentioned as category of coset bijections, defined as follows: The category is modified in case S is categorical but not strictly categorical by adding the requirement that, for each pair A ≥ B, C(A, B) contains the empty bijection. And in the case of empty composites an A − B labelled copy of the empty partial bijection with empty composites, given the appropriate labeling to avoid confusing empty partial bijections in different morphism sets.
By the nature of the coset bijections, C is a self dual category.
The following research was proposed to us by Jonathan Leech for the purpose of the author's Ph.D. dissertation in [20], and shows that not all skew lattices can determine such a category as strictly categorical skew lattices do.
In fact, ψ×ϕ needs not be the direct composite of partial functions if S is not categorical. Moreover, χ : A → C always exists as a coset bijection and always contains ψ × ϕ, which is trivial in the case that ψ × ϕ is empty.
These four examples are likely to be the minimal possible examples.

Strictly categorical skew lattices in rings
Let R = (R, +, ·) be a ring and E(R) the set of all idempotent elements in R. Set x∧y = xy and x ∨ y = x • y = x + y − xy. x • y needs not be idempotent (cf. [7]). A regular band B in a ring does not generate a skew lattice in general but the assertion is true if B satisfies a stronger identity, that is if B is in fact a normal band. If S ⊆ E(R) is closed under both · and • then (S; ·, •) is a skew lattice. Another possible choice for the operation ∨ is ∇, defined by, In general, the operation ∇ needs not be associative (conf. [3], ex 2.1). Though, ∇ is associative in the presence of normality (conf. [3], prop 2.2). By a skew lattice in a ring R we mean a set S ⊆ E(R) that is closed under both multiplication and ∇, and forms a skew lattice for the two operations. In particular, we have to make sure that ∇ is associative in S. Given a multiplicative band B in a ring R the relation between • and ∇ is given by 2 for all e, f ∈ B. Thus, a•b and a∇b coincide whenever a•b is idempotent. In the case of right-handed skew lattices the nabla operation reduces to the circle operation. Any normal multiplicative band of idempotents in a ring generates a skew lattice under multiplication and the operation ∇ with the reduct (S, ·) also being normal. The converse is however false, that is, a skew lattice whose multiplicative reduct is not normal exist (cf. [3]). Hence, skew lattices in rings need not be normal. The standard form for pure bands in matrix rings was developed by Fillmore at al. in [8] and [9]. Based on it, Cvetko-Vah described in [2] the standard form for right-handed skew lattices in M n (F ) as follows: let E 1 < · · · < E m be a maximal chain of D-classes of the skew lattice S. Then a basis for F n exists such that in this basis, for any three matrices a ∈ E i , b ∈ E j and c ∈ E k , i > j > k, a block decomposition exists such that a, b and c have block forms where a ij ∈ A, b ij , b ′ ij ∈ B and c ij ∈ C.
Proof. Consider A = E i , B = E j and C = E l and fix a k ∈ A, b k ∈ B and c k ∈ C such that a k > b k > c k presented below: so that the A-coset in B and the C-coset in B are given by Recall that all skew lattices in rings are categorical. Then, due to Proposition 9, for all b, b ′ ∈ B, (bA) ∩ (C • b ′ ) = ∅. Thus, the above equations hold. Skew lattices in rings are distributive, symmetric and categorical (cf. [11] and [13]). It is well known that ∧-distributive skew lattices are exactly the skew lattices that are simultaneously symmetric and normal for which the lattice image S/D is distributive (cf. [12]). Hence, whenever S/D is distributive, the conditions of Proposition 21 determine the skew lattices of matrices in rings that are ∧-distributive.