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Let $\mathcal{A}=(Q,\Sigma,\delta)$ be a finite deterministic complete automaton. is called k-compressible if there is a word w ∈ Σ + such that the image of the state set Q under the action of w has at most size |Q| − k, in such case the word w is called k-compressing for . A word w ∈ Σ + is k-collapsing if it is k-compressing for each k-compressible automaton...
We show that the word problem is decidable for an amalgamated free product of finite inverse semigroups (in the category of inverse semigroups). This is in contrast to a recent result of M. Sapir that shows that the word problem for amalgamated free products of finite semigroups (in the category of semigroups) is in general undecidable.
We study the lattice. C(S) of congruences of a monoid S which is the Bruck-Reilly extension of a monoid T by a homomorphism α. The inclusion, meet and join of congruences are described in terms of congruences and ideals of T. We show that C(S) can be naturally decomposed into three sublattices, corresponding (roughly speaking) to the three different types of congruences on such semigroups.
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