Skew lattice#Skew Boolean algebras

A skew lattice is a set S equipped with two associative, idempotent binary operations $\backslash wedge$ and $\backslash vee$, called meet and join, that validate the following dual pair of absorption laws

Given that $\backslash vee$ and $\backslash wedge$ are associative and idempotent, these identities are equivalent to validating the following dual pair of statements:

For over 60 years, noncommutative variations of lattices have been studied with differing motivations. For some the motivation has been an interest in the conceptual boundaries of lattice theory; for others it was a search for noncommutative forms of logic and Boolean algebra; and for others it has been the behavior of idempotents in rings. A noncommutative lattice, generally speaking, is an algebra $(S;\; \backslash wedge,\; \backslash vee)$ where $\backslash wedge$ and $\backslash vee$ are associative, idempotent binary operations connected by absorption identities guaranteeing that $\backslash wedge$ in some way dualizes $\backslash vee$. The precise identities chosen depends upon the underlying motivation, with differing choices producing distinct varieties of algebras.

Pascual Jordan, motivated by questions in quantum logic, initiated a study of noncommutative lattices in his 1949 paper, Über Nichtkommutative Verbände,Jordan, P. Uber Nichtkommutative Verbände, Arch. Math. 2 (1949), 56–59. choosing the absorption identities

He referred to those algebras satisfying them as Schrägverbände. By varying or augmenting these identities, Jordan and others obtained a number of varieties of noncommutative lattices.

Beginning with Jonathan Leech's 1989 paper, Skew lattices in rings,Leech, J, Skew lattices in rings, Algebra Universalis, 26(1989), 48-72 skew lattices as defined above have been the primary objects of study. This was aided by previous results about bands. This was especially the case for many of the basic properties.

Natural partial order and natural quasiorder

In a skew lattice $S$, the natural partial order is defined by $y\backslash leq\; x$ if $x\; \backslash wedge\; y\; =\; y\; =\; y\; \backslash wedge\; x$, or dually, $x\; \backslash vee\; y\; =\; x\; =\; y\; \backslash vee\; x$. The natural preorder on $S$ is given by $y\; \backslash preceq\; x$ if $y\; \backslash wedge\; x\backslash wedge\; y\; =\; y$ or dually $x\; \backslash vee\; y\; \backslash vee\; x\; =\; x$. While $\backslash leq$ and $\backslash preceq$ agree on lattices, $\backslash leq$ properly refines $\backslash preceq$ in the noncommutative case. The induced natural equivalence $D$ is defined by $xDy$ if $x\; \backslash preceq\; y\; \backslash preceq\; x$, that is, $x\; \backslash wedge\; y\; \backslash wedge\; x\; =\; x$

and $y\; \backslash wedge\; x\; \backslash wedge\; y\; =\; y$ or dually, $x\; \backslash vee\; y\; \backslash vee\; x\; =\; x$ and $y\; \backslash vee\; x\; \backslash vee\; y\; =\; y$. The blocks of the partition $S/D$ are

lattice ordered by $A\; >\; B$ if and only if $a\; \backslash in\; A$ and $b\; \backslash in\; B$ exist such that $a\; >\; b$. This permits us to draw Hasse diagrams of skew lattices such as the following pair:

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E.g., in the diagram on the left above, that $a$ and $b$ are $D$ related is expressed by the dashed

segment. The slanted lines reveal the natural partial order between elements of the distinct $D$-classes. The elements $1$, $c$ and $0$ form the singleton $D$-classes.

Rectangular Skew Lattices

Skew lattices consisting of a single $D$-class are called rectangular. They are characterized by the equivalent identities: $x\; \backslash wedge\; y\; \backslash wedge\; x\; =\; x$, $y\; \backslash vee\; x\; \backslash vee\; y\; =\; y$ and $x\; \backslash vee\; y\; =\; y\; \backslash wedge\; x$. Rectangular skew lattices are isomorphic to skew lattices having the following construction (and conversely): given nonempty

sets $L$ and $R$, on $L\; \backslash times\; R$ define $(x,\; y)\; \backslash vee\; (z\; ,\; w\; )\; =\; (z\; ,\; y)$ and $(x,\; y)\; \backslash wedge\; (z\; ,\; w\; )\; =\; (x,\; w\; )$. The $D$-class partition of a skew lattice $S$, as indicated in the above diagrams, is the unique partition of $S$ into its maximal rectangular subalgebras, Moreover, $D$ is a congruence with the induced quotient algebra $S/\; D$ being the maximal lattice image of $S$, thus making every skew lattice $S$ a lattice of rectangular subalgebras. This is the Clifford–McLean theorem for skew lattices, first given for bands separately by Clifford and McLean. It is also known as the first decomposition theorem for skew lattices.

Right (left) handed skew lattices and the Kimura factorization

A skew lattice is right-handed if it satisfies the identity $x\backslash wedge\; y\; \backslash wedge\; x\; =\; y\; \backslash wedge\; x$ or dually, $x\; \backslash vee\; y\; \backslash vee\; x\; =\; x\; \backslash vee\; y$.

These identities essentially assert that $x\; \backslash wedge\; y\; =\; y$ and $x\; \backslash vee\; y\; =\; x$ in each $D$-class. Every skew lattice $S$ has a unique maximal right-handed image $S/\; L$ where the congruence $L$ is defined by $xLy$ if both $x\; \backslash wedge\; y\; =\; x$ and $y\; \backslash wedge\; x\; =\; y$ (or dually, $x\; \backslash vee\; y\; =\; y$ and $y\; \backslash vee\; x\; =\; x$). Likewise a skew lattice is left-handed if $x\; \backslash wedge\; y\; =\; x$ and $x\; \backslash vee\; y\; =\; y$ in each $D$-class. Again the maximal left-handed image of a skew lattice $S$ is the image $S/\; R$ where the congruence $R$ is defined in dual fashion to $L$. Many examples of skew lattices are either right- or left-handed. In the lattice of congruences, $R\; \backslash vee\; L\; =\; D$ and $R\; \backslash cap\; L$ is the identity congruence $\backslash Delta$. The induced epimorphism $S\backslash rightarrow\; S/D$ factors through both induced epimorphisms $S\; \backslash rightarrow\; S/L$ and $S\; \backslash rightarrow\; S/R$. Setting $T\; =\; S/D$, the homomorphism $k\; :\; S\; \backslash rightarrow\; S/L\; \backslash times\; S/R$ defined by $k(x)\; =\; (L\_x,\; R\_x)$, induces an isomorphism $k*\; :\; S\backslash sim\; S/\; L\; \backslash times\; \_T\; S/R$. This is the Kimura factorization of $S$ into a fibred product of its maximal right- and left-handed images.

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Like the Clifford–McLean theorem, Kimura factorization (or the second decomposition theorem for skew lattices) was first given for regular bands (bands that satisfy the middle absorption

identity, $xyxzx\; =\; xyzx$). Indeed, both $\backslash wedge$ and $\backslash vee$ are regular band operations. The above symbols $D$, $R$ and $L$ come, of course, from basic semigroup theory.Leech, J, Recent developments in the theory of skew lattices, Semigroup Forum, 52(1996), 7-24.Leech, J, Magic squares, finite planes and simple quasilattices, Ars Combinatoria 77(2005), 75-96.Leech, J, The geometry of skew lattices, Semigroup Forum, 52(1993), 7-24.Leech, J, Normal skew lattices, Semigroup Forum, 44(1992), 1-8.Cvetko-Vah, K, Internal decompositions of skew lattices, Communications in Algebra, 35 (2007), 243-247Cvetko-Vah, K, A new proof of Spinks’ Theorem, Semigroup Forum 73 (2006), 267-272.Laslo, G and Leech, J, Green’s relations on noncommutative lattices, Acta Sci. Math. (Szeged), 68 (2002), 501-533.

Skew lattices form a variety. Rectangular skew lattices, left-handed and right-handed skew lattices all form subvarieties that are central to the basic structure theory of skew lattices. Here are several

more.

Symmetric skew lattices

A skew lattice S is symmetric if for any $x,\; y\; \backslash in\; S$ , $x\; \backslash wedge\; y\; =\; y\; \backslash wedge\; x$ if and only if $x\; \backslash vee\; y\; =\; y\; \backslash vee\; x$. Occurrences of commutation are thus unambiguous for such skew lattices, with subsets of pairwise commuting elements generating commutative subalgebras, i.e., sublattices. (This is not true for skew lattices in general.) Equational bases for this subvariety, first given by SpinksSpinks, M, Automated deduction in non-commutative lattice theory, Tech. Report 3/98, Monash U, GSCIT, 1998 are:

$x\; \backslash vee\; y\; \backslash vee\; (x\; \backslash wedge\; y)\; =\; (y\; \backslash wedge\; x)\; \backslash vee\; y\; \backslash vee\; x$ and $x\; \backslash wedge\; y\; \backslash wedge\; (x\; \backslash vee\; y)\; =\; (y\; \backslash vee\; x)\; \backslash wedge\; y\; \backslash wedge\; x$.

A lattice section of a skew lattice $S$ is a sublattice $T$ of $S$ meeting each $D$-class of $S$ at a single element. $T$ is thus an internal copy of the lattice $S/\; D$ with the composition $T\; \backslash subseteq\; S\; \backslash rightarrow\; S/D$ being an isomorphism. All symmetric skew lattices for which $|S/D|\; \backslash leq\; \backslash aleph\_0$ admit a lattice section. Symmetric or not, having a lattice section $T$ guarantees that $S$ also has internal copies of $S/\; L$ and $S/\; R$ given respectively by $T\; [R]\; =\; \backslash bigcup\_\{t\backslash in\; T\}\; R\_t$ and $T\; [L]\; =\; \backslash bigcup\_\{t\backslash in\; T\}\; L\_t$, where $R\_t$ and $Lt$ are the $R$ and $L$ congruence classes of $t$ in $T$ . Thus $T\; [\; R]\; \backslash subseteq\; S\; \backslash rightarrow\; S/L$ and $T\; [L]\; \backslash subseteq\; S\; \backslash rightarrow\; S/R$ are isomorphisms. This leads to a commuting diagram of embedding dualizing the preceding Kimura diagram.

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Cancellative skew lattices

A skew lattice is cancellative if $x\; \backslash vee\; y\; =\; x\; \backslash vee\; z$ and $x\; \backslash wedge\; y\; =\; x\; \backslash wedge\; z$ implies $y\; =\; z$ and likewise $x\; \backslash vee\; z\; =\; y\; \backslash vee\; z$ and $x\; \backslash wedge\; z\; =\; y\; \backslash wedge\; z$ implies $x\; =\; y$. Cancellatice skew lattices are symmetric and can be shown to form a variety. Unlike lattices, they need not be distributive, and conversely.

Distributive skew lattices

Distributive skew lattices are determined by the identities:

$x\; \backslash wedge\; (y\; \backslash vee\; z\; )\; \backslash wedge\; x\; =\; (x\; \backslash wedge\; y\; \backslash wedge\; x)\; \backslash vee\; (x\; \backslash wedge\; z\; \backslash wedge\; x)$ (D1)

$x\; \backslash vee\; (y\; \backslash wedge\; z\; )\; \backslash vee\; x\; =\; (x\; \backslash vee\; y\; \backslash vee\; x)\; \backslash wedge\; (x\; \backslash vee\; z\; \backslash vee\; x).$ (D'1)

Unlike lattices, (D1) and (D'1) are not equivalent in general for skew lattices, but they are for symmetric skew lattices.Spinks, M, Automated deduction in non-commutative lattice theory, Tech. Report 3/98, Monash University, Gippsland School of Computing and Information Technology, June 1998 The condition (D1) can be strengthened to

$x\; \backslash wedge\; (y\; \backslash vee\; z\; )\; \backslash wedge\; w\; =\; (x\; \backslash wedge\; y\; \backslash wedge\; w)\; \backslash vee\; (x\; \backslash wedge\; z\; \backslash wedge\; w)$ (D2)

in which case (D'1) is a consequence. A skew lattice $S$ satisfies both (D2) and its dual, $x\; \backslash vee\; (y\; \backslash wedge\; z\; )\; \backslash vee\; w\; =\; (x\; \backslash vee\; y\; \backslash vee\; w)\; \backslash wedge\; (x\; \backslash vee\; z\; \backslash vee\; w)$, if and only if it factors as the product of a distributive lattice and a rectangular skew lattice. In this latter case (D2) can be strengthened to

$x\; \backslash wedge\; (y\; \backslash vee\; z\; )\; =\; (x\; \backslash wedge\; y)\; \backslash vee\; (x\; \backslash wedge\; z\; )$ and $(y\; \backslash vee\; z\; )\; \backslash wedge\; w\; =\; (y\; \backslash wedge\; w)\; \backslash vee\; (z\; \backslash wedge\; w)$. (D3)

On its own, (D3) is equivalent to (D2) when symmetry is added. We thus have six subvarieties of skew lattices determined respectively by (D1), (D2), (D3) and their duals.

Normal skew lattices

As seen above, $\backslash wedge$ and $\backslash vee$ satisfy the identity $xyxzx\; =\; xyzx$. Bands satisfying the stronger identity, $xyzx\; =\; xzyx$, are called normal. A skew lattice is normal skew if it satisfies

$x\; \backslash wedge\; y\; \backslash wedge\; z\; \backslash wedge\; x\; =\; x\; \backslash wedge\; z\; \backslash wedge\; y\; \backslash wedge\; x.\; (N)$

For each element a in a normal skew lattice $S$, the set $a\; \backslash wedge\; S\; \backslash wedge\; a$ defined by {$a\; \backslash wedge\; x\; \backslash wedge\; a\; |\; x\; \backslash in\; S$} or equivalently {$x\; \backslash in\; S\; |\; x\backslash leq\; a$} is a sublattice of $S$, and conversely. (Thus normal skew lattices have also been called local lattices.) When both $\backslash wedge$ and $\backslash vee$ are normal, $S$ splits isomorphically into a product $T\; \backslash times\; D$ of a lattice $T$ and a rectangular skew lattice $D$, and conversely. Thus both normal skew lattices and split skew lattices form varieties. Returning to distribution, $(D2)\; =\; (D1)\; +\; (N)$ so that $(D2)$ characterizes the variety of distributive, normal skew lattices, and (D3) characterizes the variety of symmetric, distributive, normal skew lattices.

Categorical skew lattices

A skew lattice is categorical if nonempty composites of coset bijections are coset bijections. Categorical skew lattices form a variety. Skew lattices in rings and normal skew lattices are examples

of algebras in this variety. Let $a\; >\; b\; >\; c$ with $a\; \backslash in\; A$, $b\; \backslash in\; B$ and $c\; \backslash in\; C$, $\backslash varphi$ be the coset bijection from $A$ to $B$ taking $a$ to $b$, $\backslash psi$ be the coset bijection from $B$ to $C$ taking $b$ to $c$ and finally $\backslash chi$ be the coset bijection from $A$ to $C$ taking $a$ to $c$. A skew lattice $S$ is categorical if one always has the equality $\backslash psi\; \backslash circ\; \backslash varphi\; =\; \backslash chi$, i.e. , if the

composite partial bijection $\backslash psi\; \backslash circ\; \backslash varphi$ if nonempty is a coset bijection from a $C$ -coset of $A$ to an $A$-coset

of $C$ . That is $(A\; \backslash wedge\; b\; \backslash wedge\; A)\; \backslash cap\; (C\; \backslash vee\; b\; \backslash vee\; C\; )\; =\; (C\; \backslash vee\; a\; \backslash vee\; C\; )\; \backslash wedge\; b\; \backslash wedge\; (C\; \backslash vee\; a\; \backslash vee\; C\; )\; =\; (A\; \backslash wedge\; c\; \backslash wedge\; A)\; \backslash vee\; b\; \backslash vee\; (A\; \backslash wedge\; c\; \backslash wedge\; A)$.

All distributive skew lattices are categorical. Though symmetric skew lattices might not be. In a sense they reveal the independence between the properties of symmetry and distributivity.Spinks, M, On middle distributivity for skew lattices, Semigroup Forum 61 (2000), 341-345.Cvetko-Vah, Karin ; Kinyon, M. ; Leech, J. ; Spinks, M. [https://cs.du.edu/~mathfiles/preprints/nsm-math-preprint-0813.pdf Cancellation in skew Lattices]. Order 28 (2011), 9-32.

A zero element in a skew lattice S is an element 0 of S such that for all $x\; \backslash in\; S,$ $0\; \backslash wedge\; x\; =\; 0\; =\; x\; \backslash wedge\; 0$ or, dually, $0\; \backslash vee\; x\; =\; x\; =\; x\; \backslash vee\; 0.$ (0)

A Boolean skew lattice is a symmetric, distributive normal skew lattice with 0, $(S\; ;\; \backslash vee,\; \backslash wedge,\; 0),$ such that $a\; \backslash wedge\; S\; \backslash wedge\; a$ is a Boolean lattice for each $a\; \backslash in\; S.$ Given such skew lattice S, a difference operator \ is defined by x \ y = $x\; -\; x\; \backslash wedge\; y\; \backslash wedge\; x$ where the latter is evaluated in the Boolean lattice $x\; \backslash wedge\; S\; \backslash wedge\; x.$ In the presence of (D3) and (0), \ is characterized by the identities:

$y\; \backslash wedge\; x\; \backslash setminus\; y\; =\; 0\; =\; x\; \backslash setminus\; y\; \backslash wedge\; y$ and $(x\; \backslash wedge\; y\; \backslash wedge\; x)\; \backslash vee\; x\; \backslash setminus\; y\; =\; x\; =\; x\; \backslash setminus\; y\; \backslash vee\; (x\; \backslash wedge\; y\; \backslash wedge\; x).$ (S B)

One thus has a variety of skew Boolean algebras $(S\; ;\; \backslash vee,\; \backslash wedge,\; \backslash ,\; 0)$ characterized by identities (D3), (0) and (S B). A primitive skew Boolean algebra consists of 0 and a single non-0 D-class. Thus it is the result of adjoining a 0 to a rectangular skew lattice D via (0) with $x\; \backslash setminus\; y\; =\; x$, if $y\; =\; 0$

and $0$ otherwise. Every skew Boolean algebra is a subdirect product of primitive algebras. Skew Boolean algebras play an important role in the study of discriminator varieties and other generalizations in universal algebra of Boolean behavior.Bignall, R. J., Quasiprimal Varieties and Components of Universal Algebras, Dissertation, The Flinders University of South Australia, 1976.Bignall, R J, A non-commutative multiple-valued logic, Proc. 21st International Symposium on Multiple-valued Logic, 1991, IEEE Computer Soc. Press, 49-54.Bignall, R J and J Leech, Skew Boolean algebras and discriminator varieties, Algebra Universalis, 33(1995), 387-398.Bignall, R J and M Spinks, Propositional skew Boolean logic, Proc. 26th International Symposium on Multiple-valued Logic, 1996, IEEE Computer Soc. Press, 43-48.Bignall, R J and M Spinks, [https://pdfs.semanticscholar.org/9db0/3abeb9c20190523db14a8486f325eede46f9.pdf Implicative BCS-algebra subreducts of skew Boolean algebras], Scientiae Mathematicae Japonicae, 58 (2003), 629-638.Bignall, R J and M Spinks, On binary discriminator varieties (I): Implicative BCS-algebras, International Journal of Algebra and Computation, to appear.Cornish, W H, Boolean skew algebras, Acta Math. Acad. Sci. Hung., 36 (1980), 281-291.Leech, J, Skew Boolean algebras, Algebra Universalis, 27(1990), 497-506.Leech and Spinks, Skew Boolean algebras generated from generalized Boolean algebras, Algebra Universalis 58 (2008), 287-302, 307-311.Spinks, M, Contributions to the Theory of Pre-BCK Algebras, Monash University Dissertation, 2002.Spinks, M and R Veroff, [https://www.cs.unm.edu/~veroff/DOCS/sbpc.pdf Axiomatizing the skew Boolean propositional calculus], J. Automated Reasoning, 37 (2006), 3-20.

Let $A$ be a ring and let $E(A)$ denote the set of all idempotents in $A$. For all $x,\; y\; \backslash in\; A$ set $x\; \backslash wedge\; y\; =\; xy$ and $x\; \backslash vee\; y\; =\; x\; +\; y\; -\; xy$.

Clearly $\backslash wedge$ but also $\backslash vee$ is associative. If a subset $S\; \backslash subseteq\; E(A)$ is closed under $\backslash wedge$ and $\backslash vee$, then $(S,\; \backslash wedge,\; \backslash vee)$ is a distributive, cancellative skew lattice. To find such skew lattices in $E(A)$ one looks at bands in $E(A)$, especially the ones that are maximal with respect to some constraint. In fact, every multiplicative band in $()$ that is maximal with respect to being right regular (= ) is also closed under $\backslash vee$ and so forms a right-handed skew lattice. In general, every right regular band in $E(A)$ generates a right-handed skew lattice in $E(A)$. Dual remarks also hold for left regular bands (bands satisfying the identity $xyx\; =\; xy$) in $E(A)$. Maximal regular bands need not to be closed under $\backslash vee$ as defined; counterexamples are easily found using multiplicative rectangular bands. These cases are closed, however, under the cubic variant of $\backslash vee$ defined by $x\; \backslash nabla\; y\; =\; x\; +\; y\; +\; yx\; -\; xyx\; -\; yxy$ since in these cases $x\; \backslash nabla\; y$ reduces to $yx$ to give the dual rectangular band. By replacing the condition of regularity by normality $(xyz\; w\; =\; xz\; yw)$, every maximal normal multiplicative band $S$ in $E(A)$ is also closed under $\backslash nabla$ with $(S\; ;\; \backslash wedge,\; \backslash vee,\; /,\; 0)$, where $x/y\; =\; x\; -\; xyx$, forms a Boolean skew lattice. When $E(A)$ itself is closed under multiplication, then it is a normal band and thus forms a Boolean skew lattice. In fact, any skew Boolean algebra can be embedded into such an algebra.Cvetko-Vah, K, Skew lattices in matrix rings, Algebra Universalis 53 (2005), 471-479. When A has a multiplicative identity $1$, the condition that $E(A)$ is multiplicatively closed is well known to imply that $E(A)$ forms a Boolean algebra. Skew lattices in rings continue to be a good source of examples and motivation.Cvetko-Vah, K, Pure skew lattices in rings, Semigroup Forum 68 (2004), 268-279.Cvetko-Vah, K, Pure ∇-bands, Semigroup Forum 71 (2005), 93-101.Cvetko-Vah, K, Skew lattices in rings, Dissertation, University of Ljubljana, 2005.Cvetko-Vah, K and J Leech, Associativity of the ∇-operation on bands in rings, Semigroup Forum 76 (2008), 32-50

Skew lattices consisting of exactly two D-classes are called primitive skew lattices. Given such a skew lattice $S$ with $D$-classes $A\; >\; B$ in $S/D$, then for any $a\; \backslash in\; A$ and $b\; \backslash in\; B$, the subsets

$A\; \backslash wedge\; b\; \backslash wedge\; A\; =${$u\; \backslash wedge\; b\; \backslash wedge\; u\; :\; u\; \backslash in\; A$} $\backslash subseteq\; B$ and $B\; \backslash vee\; a\; \backslash vee\; B\; =$ {$v\; \backslash vee\; a\; \backslash vee\; v\; :\; v\; \backslash in\; B$} $\backslash subseteq\; A$

are called, respectively, cosets of A in B and cosets of B in A. These cosets partition B and A with $b\; \backslash in\; A\backslash wedge\; b\backslash wedge\; A$ and $a\; \backslash in\; B\backslash wedge\; a\backslash wedge\; B$. Cosets are always rectangular subalgebras in their $D$-classes. What is more, the partial order $\backslash geq$ induces a coset bijection $\backslash varphi\; :\; B\; \backslash vee\; a\; \backslash vee\; B\; \backslash rightarrow\; A\; \backslash wedge\; b\; \backslash wedge\; A$ defined by:

$\backslash phi\; (x)\; =\; y$ iff $x\; >\; y$, for $x\; \backslash in\; B\; \backslash vee\; a\; \backslash vee\; B$ and $y\; \backslash in\; A\; \backslash wedge\; b\; \backslash wedge\; A$.

Collectively, coset bijections describe $\backslash geq$ between the subsets $A$ and $B$. They also determine $\backslash vee$ and $\backslash wedge$ for pairs of elements from distinct $D$-classes. Indeed, given $a\; \backslash in\; A$ and $b\; \backslash in\; B$, let $\backslash varphi$ be the

cost bijection between the cosets $B\backslash vee\; a\backslash vee\; B$ in $A$ and $A\; \backslash wedge\; b\; \backslash wedge\; A$ in $B$. Then:

$a\backslash vee\; b\; =\; a\backslash vee\; \backslash varphi\; -1(b),\; b\backslash vee\; a\; =\; \backslash varphi\; -1(b)\backslash vee\; a$ and $a\backslash wedge\; b\; =\; \backslash varphi(a)\backslash wedge\; b\; ,\; b\backslash wedge\; a\; =\; b\backslash wedge\; \backslash varphi\; (a)$.

In general, given $a,\; c\; \backslash in\; A$ and $b,\; d\; \backslash in\; B$ with $a\; >\; b$ and $c\; >\; d$, then $a,\; c$ belong to a common $B$- coset in $A$ and $b,\; d$ belong to a common $A$-coset in $B$ if and only if $a\; >\; b\; //\; c\; >\; d$. Thus each coset bijection is, in some sense, a maximal collection of mutually parallel pairs $a\; >\; b$.

Every primitive skew lattice $S$ factors as the fibred product of its maximal left and right- handed primitive images $S/R\; \backslash times\_2\; S/L$. Right-handed primitive skew lattices are constructed as follows. Let $A\; =\; \backslash cup\; \_i\; A\_i$ and $B\; =\; \backslash cup\; \_j\; B\_j$ be partitions of disjoint nonempty sets $A$ and $B$, where all $A\_i$ and $B\_j$ share a common size. For each pair $i,\; j$ pick a fixed bijection $\backslash varphi\_i,j$ from $A\_i$ onto $B\_j$. On $A$ and $B$ separately set $x\backslash wedge\; y\; =\; y$ and $x\backslash vee\; y\; =\; x$; but given $a\; \backslash in\; A$ and $b\; \backslash in\; B$, set

$a\; \backslash vee\; b\; =\; a,\; b\; \backslash vee\; a\; =\; a\text{'},\; a\; \backslash wedge\; b\; =\; b$ and $b\; \backslash wedge\; a\; =\; b\text{'}$

where $\backslash varphi\_\{i,j\}(a\text{'})\; =\; b$ and $\backslash varphi\_\{i,j\}(a)\; =\; b\text{'}$ with $a\text{'}$ belonging to the cell $A\_i$ of $a$ and $b\text{'}$ belonging to the cell $B\_j$ of $b$. The various $\backslash varphi\; i,j$ are the coset bijections. This is illustrated in the following partial Hasse diagram where $|A\_i|\; =\; |B\_j|\; =\; 2$ and the arrows indicate the $\backslash varphi\_\{i,j\}$ -outputs and $\backslash geq$ from $A$ and $B$.

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One constructs left-handed primitive skew lattices in dual fashion. All right [left] handed primitive skew lattices can be constructed in this fashion.

A nonrectangular skew lattice $S$ is covered by its maximal primitive skew lattices: given comparable $D$-classes $A\; >\; B$ in $S/D$, $A\; \backslash cup\; B$ forms a maximal primitive subalgebra of $S$ and every $D$-class in $S$ lies in such a subalgebra. The coset structures on these primitive subalgebras combine to determine the outcomes $x\backslash vee\; y$ and $x\backslash wedge\; y$ at least when $x$ and $y$ are comparable under $\backslash preceq$. It turns out that $x\backslash vee\; y$ and $x\backslash wedge\; y$ are determined in general by cosets and their bijections, although in

a slightly less direct manner than the $\backslash preceq$-comparable case. In particular, given two incomparable D-classes A and B with join D-class J and meet D-class $M$ in $S/D$, interesting connections arise between the two coset decompositions of J (or M) with respect to A and B.

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Thus a skew lattice may be viewed as a coset atlas of rectangular skew lattices placed on the vertices of a lattice and coset bijections between them, the latter seen as partial isomorphisms

between the rectangular algebras with each coset bijection determining a corresponding pair of cosets. This perspective gives, in essence, the Hasse diagram of the skew lattice, which is easily

drawn in cases of relatively small order. (See the diagrams in Section 3 above.) Given a chain of D-classes $A\; >\; B\; >\; C$ in $S/D$, one has three sets of coset bijections: from A to B, from B to C and from A to C. In general, given coset bijections $\backslash varphi:\; A\; \backslash rightarrow\; B$ and $\backslash psi:\; B\; \backslash rightarrow\; C$, the composition of partial bijections $\backslash psi\; \backslash varphi$ could be empty. If it is not, then a unique coset bijection $\backslash chi:\; A\; \backslash rightarrow\; C$ exists such that $\backslash psi\; \backslash varphi\; \backslash subseteq\; \backslash chi$. (Again, $\backslash chi$ is a bijection between a pair of cosets in $A$ and $C$.) This inclusion can be strict. It is always an equality (given $\backslash psi\; \backslash varphi\; \backslash neq\; \backslash empty$) on a given skew lattice S precisely when S is categorical. In this case, by including the identity maps on each rectangular D-class and adjoining empty bijections between properly comparable D-classes, one has a category of rectangular algebras and coset bijections between them. The simple examples in Section 3 are categorical.

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