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Directed graphs and combinatorial properties of semigroups. (English) Zbl 1005.20043
Let \(D\) be a fixed finite graph. The graph \(G=(V,E)\) is said to be \(D\)-saturated if for every infinite subset \(W\) of \(V\) there exists a subgraph of \(G\) isomorphic with \(D\) having all vertices in \(W\).
Let \(S\) be a semigroup. The authors associate to \(S\) the following three graphs: \(\text{Pow}(S)\), \(\text{Div}(S)\), \(\text{Ann}(S)\) for which \(S\) is the vertex set (in the construction of \(\text{Div}(S)\) \(S\) is assumed to be a semigroup with \(0\)). The set of edges is defined as follows. If \(u,v\in S\) and \(u\neq v\) then:
\((u,v)\) is a edge for \(\text{Pow}(s)\iff v\) is a power of \(u\).
\((u,v)\) is a edge for \(\text{Div}(S)\iff u\) divides \(v\),
\((u,v)\) is a edge for \(\text{Ann}(S)\iff uv=0\).
The purpose of this paper is to characterize the commutative semigroups \(S\) for which the graphs \(\text{Pow}(S)\) or \(\text{Div}(S)\) or \(\text{Ann}(S)\) are \(D\)-saturated.

MSC:
20M14 Commutative semigroups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C20 Directed graphs (digraphs), tournaments
20M05 Free semigroups, generators and relations, word problems
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