skew_lattices - MathStructures

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=====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: 
$h(x \vee y)=h(x) \vee h(y)$,
$h(x \wedge y)=h(x) \wedge h(y)$,

====Examples====
Example 1: 

====Basic results====


====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  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====

[(Leech1989> 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.
 
)]\end{document}
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