Geodesic
Presentations and word problem for strong semilattices of semigroups
Algebra and discrete mathematics, no. 4 (2005), pp. 28-35

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Let $I$ be a semilattice, and $S_i(i\in I)$ be a family of disjoint semigroups. Then we prove that the strong semilattice $S=\mathcal{S} [I,S_i,\phi_{j,i}]$ of semigroups $S_i$ with homomorphisms $\phi _{j,i}:S_j\rightarrow S_i$ $(j\geq i)$ is finitely presented if and only if $I$ is finite and each $S_i$ $(i\in I)$ is finitely presented. Moreover, for a finite semilattice $I$, $S$ has a soluble word problem if and only if each $S_i$ $(i\in I)$ has a soluble word problem. Finally, we give an example of non-automatic semigroup which has a soluble word problem.
Keywords: Semigroup presentations, strong semilattices of semigroups, word problems. 
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     author = {Gonca Ayik and Hayrullah Ayik and Yu. \"Unl\"u},
     title = {Presentations and word problem for strong semilattices of semigroups},
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     pages = {28--35},
     publisher = {mathdoc},
     number = {4},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2005_4_a2/}
}
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Gonca Ayik; Hayrullah Ayik; Yu. Ünlü. Presentations and word problem for strong semilattices of semigroups. Algebra and discrete mathematics, no. 4 (2005), pp. 28-35. http://geodesic.mathdoc.fr/item/ADM_2005_4_a2/