O-direct union

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2020 Mathematics Subject Classification: Primary: 20M10 [MSN][ZBL]

of semi-groups with zero

The semi-group obtained from the given family $\{S_\alpha\}$ of semi-groups with zero, pairwise intersecting only at the zero element, by specifying on $\bigcup_\alpha S_\alpha$ the multiplication operation that coincides with the original operation on each semi-group $S_\alpha$ and is such that $S_\alpha S_\beta = 0$ for different $\alpha, \beta$. The $O$-direct union is also called the orthogonal sum. A number of types of semi-groups can be described by decomposing them in an $O$-direct union of known semi-groups (cf., e.g., Maximal ideal; Minimal ideal; Regular semi-group).


[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967)
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O-direct union. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article