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Homesemilattice decomposition of a semigroup

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# 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.

## Mathematics Subject Classification

20M10*no label found*

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