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

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

Superclasses

References


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