# Differences

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semigroups_with_zero [2010/07/29 15:46] (current)
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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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