semilattices_with_identity

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=====Semilattices with identity=====

Abbreviation: **Slat$_1$**
====Definition====
A \emph{semilattice with identity} is a structure $\mathbf{S}=\langle S,\cdot,1\rangle$ of type $\langle 2,0\rangle $ such that


$\langle S,\cdot\rangle$ is a [[semilattices]]


$1$ is an indentity for $\cdot$:  $x\cdot 1=x$

==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices with identity. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 

$h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable in PTIME |
^[[Quasiequational theory]]  |decidable |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |yes |
^[[Residual size]]  |2 |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |no |
^[[Congruence meet-semidistributive]]  |yes |
^[[Congruence n-permutable]]  |no |
^[[Congruence regular]]  |no |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  |yes |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  |yes |
^[[Strong amalgamation property]]  |yes |
^[[Epimorphisms are surjective]]  |yes |
====Finite members====

$\begin{array}{lr}
f(1)= &1\quad\\
f(2)= &1\quad\\
f(3)= &1\quad\\
f(4)= &2\quad\\
f(5)= &5\quad\\
f(6)= &15\\
\end{array}$

====Subclasses====
[[Semilattices with identity and zero]] 

====Superclasses====
[[Semilattices]] 


====References====

[(Ln19xx>
)]

Trace: • idempotent_semirings_with_zero • division_rings • pseudocomplemented_distributive_lattices • boolean_algebras • distributive_lattices_with_operators • commutative_lattice-ordered_rings • finite_monoids • bands