Difference between revisions of "Separable semi-group"
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| − | A [[Semi-group|semi-group]] in which $x^2=xy=y^2$ for elements $x$, $y$ implies that $x=y$. If a semi-group $S$ has a partition into sub-semi-groups that satisfy the cancellation law, then $S$ is separable. For commutative semi-groups the converse holds; moreover, any commutative separable semi-group can be expanded into a [[Band of semi-groups|band of semi-groups]] (trivially into a semi-lattice) with the cancellation law. A commutative semi-group is separable if and only if it can be imbedded in a [[ | + | A [[Semi-group|semi-group]] in which $x^2=xy=y^2$ for elements $x$, $y$ implies that $x=y$. If a semi-group $S$ has a partition into sub-semi-groups that satisfy the [[cancellation law]], then $S$ is separable. For commutative semi-groups the converse holds; moreover, any commutative separable semi-group can be expanded into a [[Band of semi-groups|band of semi-groups]] (trivially into a semi-lattice) with the cancellation law. A commutative semi-group is separable if and only if it can be imbedded in a [[Clifford semi-group]]. A periodic semi-group is separable if and only if it is a Clifford semi-group. A commutative semi-group $S$ is separable if and only if its characters separate the elements of $S$. |
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , '''1''' , Amer. Math. Soc. (1961)</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , '''1''' , Amer. Math. Soc. (1961) {{ZBL|0111.03403}}</TD></TR></table> |
Latest revision as of 17:58, 22 December 2023
A semi-group in which $x^2=xy=y^2$ for elements $x$, $y$ implies that $x=y$. If a semi-group $S$ has a partition into sub-semi-groups that satisfy the cancellation law, then $S$ is separable. For commutative semi-groups the converse holds; moreover, any commutative separable semi-group can be expanded into a band of semi-groups (trivially into a semi-lattice) with the cancellation law. A commutative semi-group is separable if and only if it can be imbedded in a Clifford semi-group. A periodic semi-group is separable if and only if it is a Clifford semi-group. A commutative semi-group $S$ is separable if and only if its characters separate the elements of $S$.
References
| [1] | A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961) Zbl 0111.03403 |
How to Cite This Entry:
Separable semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_semi-group&oldid=31737
Separable semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_semi-group&oldid=31737
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article