## Skew lattices

Abbreviation: SkLat

### Definition

A \emph{skew lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge\rangle,$ of type $\langle 2,2\rangle$ such that

$\langle A,\vee\rangle$ is a band,

$\langle A,\wedge\rangle$ is a band,

and the following absorption laws hold: $x\wedge (x\vee y)=x=x\vee (x\wedge y)$, $(x\vee y)\wedge y=y=(x\wedge y)\vee y$.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be skew lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$,

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &3\\ f(3)= &7\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### Subclasses

[[Lattices]] expanded type
[[Rectangular_bands]] expanded type

### Superclasses

[[Semigroups]] reduced type

### References

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1) Leech, J., Skew lattices in rings, Alg. Universalis 26 (1989), 48–72. [(Leech1993> Leech, J., The geometric structure of skew lattices, Trans. Amer. Math. Soc. 35 (1993), 823–842. [(Leech1996> Leech, J., Recent developments in the theory of skew lattices, Semigroup Forum 52 (1996), 7–24.