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Orthogonal sums of \(0\)-\(\sigma\)-simple semigroups. (English) Zbl 0849.20039

A semigroup \(S\) with zero \(0\) is an orthogonal sum of semigroups \(S_\alpha\) (\(\alpha\in A\)), if \(S_\alpha\neq 0\) for all \(\alpha\in A\), \(S=\bigcup_{\alpha\in A}S_\alpha\) and \(S_\alpha\cap S_\beta=S_\alpha\cdot S_\beta=0\) for all \(\alpha,\beta\in A\), \(\alpha\neq\beta\). In this case the family \(\{S_\alpha\mid\alpha\in A\}\) is called an orthogonal decomposition of \(S\), and \(S_\alpha\) are orthogonal summands.
Orthogonal decompositions of semigroups into summands of some types (called \(0\)-\(\sigma\)-simple, \(0\)-\(\sigma_n\)-simple, \(0\)-Archimedean) are characterized. These results are generalizations of corresponding results concerning orthogonal decompositions into (completely) \(0\)-simple summands.

MSC:

20M10 General structure theory for semigroups
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