kleene_lattices - MathStructures

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=====Kleene lattices=====

Abbreviation: **KLat**


====Definition====
A \emph{Kleene lattice} is a structure $\mathbf{A}=\langle A,\vee
,\wedge ,0,\cdot ,1,^{\ast }\rangle $ of type $\langle
2,2,0,2,0,1\rangle $ such that


$\langle A,\vee ,0,\cdot ,1,^{\ast }\rangle $ is a Kleene algebra


$\langle A,\vee ,\wedge \rangle $ is a lattice


==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be Kleene lattices. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a
homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ 
\mbox{and}  h(x\cdot y)=h(x)\cdot h(y)$, $h(x^{\ast
})=h(x)^{\ast }$, $h(0)=0$, $h(1)=1$


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


====Basic results====


====Properties====
^[[Classtype]]  |Quasivariety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  |Undecidable |
^[[First-order theory]]  |Undecidable |
^[[Locally finite]]  |No |
^[[Residual size]]  |Unbounded |
^[[Congruence distributive]]  |Yes |
^[[Congruence modular]]  |Yes |
^[[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====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &3\\
f(4)= &16\\
f(5)= &149\\
f(6)= &1488\\
\end{array}$


====Subclasses====
[[Action lattices]] 


====Superclasses====
[[Kleene algebras]] 

[[Multiplicative lattices]] 


====References====

Trace: • finite_commutative_semigroups • cancellative_partial_monoids • ordered_rings • left_neofield • commutative_lattice-ordered_semigroups • commutative_residuated_partially_ordered_semigroups • semifields • partially_ordered_monoids • partial_monoids • cancellative_semigroups