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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>
 +)]