View source for O-direct union

{{MSC|20M10}}

''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]]).

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''2''' , Amer. Math. Soc.  (1967)</TD></TR>
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