| Semigroup |
Articles about Semigroup |
Information AboutSemigroup |
| CATEGORIES ABOUT SEMIGROUP | |
| abstract algebra | |
| semigroup theory | |
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Juxtaposition suffices to denote the semigroup operation. That is, ''xy'' denotes the result of applying the semigroup operation to the ordered pair (''x'', ''y''). A semigroup with an Identity Element is a Monoid . Any semigroup ''S'' may be turned into a monoid simply by adjoining an element ''e'' not in ''S'' and defining ''es'' = ''s'' = ''se'' for all ''s'' ∈ ''S'' ∪ {e}. Some require that a semigroup have an Identity Element , which would render semigroups identical to Monoid s. Moreover, Not all agree that ''S'' should be Nonempty . This entry assumes that a semigroup may be empty, and need not have an identity. EXAMPLES OF SEMIGROUPS
STRUCTURE OF SEMIGROUPS This section sets out concepts useful for understanding the structure of semigroups. Two semigroups ''S'' and ''T'' are said to be Isomorphic if there is a Bijection ''f'' : ''S'' ↔ ''T'' with the property that, for any elements ''a'', ''b'' in ''S'', ''f''(''ab'') = ''f''(''a'')''f''(''b''). In this case, ''T'' and ''S'' are also isomorphic, and for the purposes of semigroup theory, the two semigroups are identical. |
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