=====Cancellative commutative monoids===== Abbreviation: **CanCMon** ====Definition==== A \emph{cancellative commutative monoid} is a [[cancellative monoid]] $\mathbf{M}=\langle M,\cdot ,e\rangle $ such that $\cdot $ is commutative: $x\cdot y=y\cdot x$ ==Morphisms== Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative commutative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:Marrow N$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$ ====Examples==== Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero. ====Basic results==== All commutative free monoids are cancellative. All finite commutative (left or right) cancellative monoids are reducts of [[abelian groups]]. ====Properties==== ^[[Classtype]] |quasivariety | ^[[Equational theory]] | | ^[[Quasiequational theory]] | | ^[[First-order theory]] |undecidable | ^[[Locally finite]] |no | ^[[Residual size]] |unbounded | ^[[Congruence distributive]] |no | ^[[Congruence modular]] | | ^[[Congruence n-permutable]] | | ^[[Congruence regular]] | | ^[[Congruence uniform]] | | ^[[Congruence extension property]] | | ^[[Definable principal congruences]] | | ^[[Equationally def. pr. cong.]] | | ^[[Amalgamation property]] | | ^[[Strong amalgamation property]] | | ^[[Epimorphisms are surjective]] | | ====Finite members==== $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &1\\ f(7)= &1\\ \end{array}$ ====Subclasses==== [[Abelian groups]] [[Cancellative commutative residuated lattices]] ====Superclasses==== [[Cancellative commutative semigroups]] [[Cancellative monoids]] [[Commutative monoids]] ====References==== [(Ln19xx> )]