Difference between revisions of "Right group"
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| − | A semi-group which is right simple (cf. [[Simple semi-group|Simple semi-group]]) and satisfies the left [[cancellation law]]. Every right group is a [[completely-simple semi-group]]. The property that a semi-group $S$ is a right group is equivalent to any of the following conditions: a) $S$ is right simple and contains an idempotent element; b) $S$ is regular (cf. [[Regular semi-group]]) and satisfies the left cancellation law; c) $S$ can be partitioned into left ideals which are (necessarily isomorphic) groups; and d) $S$ is the direct product of a group and a right zero semi-group (cf. [[Idempotents, semi-group of]). The notion of a left group is similar to that of a right group. Only groups are simultaneously right groups and left groups. Every completely-simple semi-group can be partitioned into right (left) ideals which are (necessarily isomorphic) right (left) groups. | + | A semi-group which is right simple (cf. [[Simple semi-group|Simple semi-group]]) and satisfies the left [[cancellation law]]. Every right group is a [[completely-simple semi-group]]. The property that a semi-group $S$ is a right group is equivalent to any of the following conditions: a) $S$ is right simple and contains an idempotent element; b) $S$ is regular (cf. [[Regular semi-group]]) and satisfies the left cancellation law; c) $S$ can be partitioned into left ideals which are (necessarily isomorphic) groups; and d) $S$ is the direct product of a group and a right zero semi-group (cf. [[Idempotents, semi-group of]]). The notion of a left group is similar to that of a right group. Only groups are simultaneously right groups and left groups. Every completely-simple semi-group can be partitioned into right (left) ideals which are (necessarily isomorphic) right (left) groups. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR></table> | ||
Latest revision as of 17:09, 8 January 2021
A semi-group which is right simple (cf. Simple semi-group) and satisfies the left cancellation law. Every right group is a completely-simple semi-group. The property that a semi-group $S$ is a right group is equivalent to any of the following conditions: a) $S$ is right simple and contains an idempotent element; b) $S$ is regular (cf. Regular semi-group) and satisfies the left cancellation law; c) $S$ can be partitioned into left ideals which are (necessarily isomorphic) groups; and d) $S$ is the direct product of a group and a right zero semi-group (cf. Idempotents, semi-group of). The notion of a left group is similar to that of a right group. Only groups are simultaneously right groups and left groups. Every completely-simple semi-group can be partitioned into right (left) ideals which are (necessarily isomorphic) right (left) groups.
References
| [1] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) |
Right group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Right_group&oldid=35784