Skew lattices

Abbreviation: SkLat

Definition

A 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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