Difference between revisions of "O-direct union"
From Encyclopedia of Mathematics
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''of semi-groups with zero'' | ''of semi-groups with zero'' | ||
| − | The [[ | + | 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==== | ====References==== | ||
| − | <table><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></table> | + | <table> |
| + | <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> | ||
| + | </table> | ||
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| + | {{TEX|done}} | ||
Latest revision as of 17:24, 19 June 2016
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).
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=11404
O-direct union. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=O-direct_union&oldid=11404
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article