A note on the strongly compatible tolerances on an arbitrary semigroup. (English) Zbl 0547.20056

A strongly compatible tolerance \(\xi\) on a semigroup S is defined similarly as a congruence, only the requirement of transitivity is omitted. A non-empty subset B of S is called a block of \(\xi\) if and only if \(B\times B\subseteq\xi \) and for each C such that \(B\subseteq C\) and \(C\times C\subseteq\xi \) the equality \(B=C\) holds. The symbol \({\mathcal H}\) denotes Green’s relation. The main result is the following theorem: Let S be an arbitrary semigroup. A strongly compatible tolerance \(\xi\) on S is an idempotent separating congruence on S if (i) \(\xi \subseteq {\mathcal H}\) and (ii) for the blocks \(\{B_ i\}_{i\in I}\) of \(\xi\) either \(B_ i\cap B_ j=\emptyset\) (\(i\neq j)\), or \(B_ i\cap B_ j\) (\(i\neq j)\) contains an idempotent.
Reviewer: B.Zelinka


20M10 General structure theory for semigroups
20M15 Mappings of semigroups
08A30 Subalgebras, congruence relations
Full Text: DOI EuDML


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