# O-direct union

From Encyclopedia of Mathematics

2010 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).

#### References

[1] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967) |

**How to Cite This Entry:**

O-direct union.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=O-direct_union&oldid=38991

This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article