×

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

MSC:

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

References:

[1] Chajda I.: Tolerance Hamiltonian varieties of algebras. Acta Sci. Maths. 44 (1982) 13-16. · Zbl 0488.08001
[2] Howie J. M.: An Introduction to Semigroup Theory. Academic Press, New York (1976). · Zbl 0355.20056
[3] Kumaresan K.: A note on idempotent separating congruence on a regular semigroup. · Zbl 0548.20046
[4] Zelinka B.: Tolerances in algebraic structures. Czech. Math. J. 20 (1970) 179- 183. · Zbl 0197.01002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.