semigroups_with_zero - MathStructures

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=====Semigroups with zero=====

Abbreviation: **Sgrp$_0$**
====Definition====
A \emph{semigroup with zero} is a structure $\mathbf{S}=\langle S,\cdot,0\rangle$ of type $\langle 2,0\rangle $ such that


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


$0$ is a zero for $\cdot$:  $x\cdot 0=0$, $0\cdot x=0$

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

$h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
====Examples====
Example 1: 

====Basic results====

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

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

====Subclasses====
[[Commutative semigroups with zero]] 

====Superclasses====
[[Semigroups]] 


====References====

[(Ln19xx>
)]

Trace: • commutative_ordered_semigroups • allegories • residuated_lattices

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