# Differences

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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> | ||

+ | )] |

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