# semilattice decomposition of a semigroup

A semigroup $S$ has a semilattice decomposition if we can write $S=\bigcup_{\gamma\in\Gamma}S_{\gamma}$ as a disjoint union of subsemigroups, indexed by elements of a semilattice $\Gamma$, with the additional condition that $x\in S_{\alpha}$ and $y\in S_{\beta}$ implies $xy\in S_{\alpha\beta}$.

Semilattice decompositions arise from homomorphims of semigroups onto semilattices. If $\phi\colon S\to\Gamma$ is a surjective homomorphism, then it is easy to see that we get a semilattice decomposition by putting $S_{\gamma}=\phi^{-1}(\gamma)$ for each $\gamma\in\Gamma$. Conversely, every semilattice decomposition defines a map from $S$ to the indexing set $\Gamma$ which is easily seen to be a homomorphism.

A third way to look at semilattice decompositions is to consider the congruence $\rho$ defined by the homomorphism $\phi\colon S\to\Gamma$. Because $\Gamma$ is a semilattice, $\phi(x^{2})=\phi(x)$ for all $x$, and so $\rho$ satisfies the constraint that $x\,\rho\,x^{2}$ for all $x\in S$. Also, $\phi(xy)=\phi(yx)$ so that $xy\,\rho\,yx$ for all $x,y\in S$. A congruence $\rho$ which satisfies these two conditions is called a semilattice congruence.

Conversely, a semilattice congruence $\rho$ on $S$ gives rise to a homomorphism from $S$ to a semilattice $S/\rho$. The $\rho$-classes are the components of the decomposition.

Title semilattice decomposition of a semigroup SemilatticeDecompositionOfASemigroup 2013-03-22 13:07:09 2013-03-22 13:07:09 mclase (549) mclase (549) 6 mclase (549) Definition msc 20M10 semilattice congruence