Empty semigroup: Difference between revisions
No edit summary |
I have a problem with this title: "zero element" should mean an element that behaves like the number zero. But this seems to mean zero is how many elements there are. |
||
| Line 1: | Line 1: | ||
In [[mathematics]], a '''semigroup with zero |
In [[mathematics]], a '''semigroup with zero elements''' (the '''empty semigroup''') is a [[semigroup]] in which the [[underlying set]] is the [[empty set]]. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a ''non-empty'' set together with an associative binary operation.<ref>A H Clifford, G B Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). [[American Mathematical Society]]. ISBN 978-0821802724 </ref><ref>{{cite book|last=J M Howie|title=An Introduction to Semigroup Theory|publisher=Academic Press|date=1976|series=L.M.S.Monographs|volume=7}} pp.2-3</ref> However not all authors insist on the underlying set of a semigroup being non-empty.<ref>P A Grillet (1995). ''Semigroups''. [[CRC Press]]. ISBN 978-0824796624 pp.3-4 </ref> One can logically define a semigroup in which the underlying set ''S'' is empty. The binary operation in the semigroup is the empty function from ''S'' × ''S'' to ''S''. This operation vacuously satisfies the associativity property. The postulation of the existence of the empty semigroup gives more generality for certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup ''T'' is a subsemigroup of ''T'' becomes valid even when the intersection is empty. |
||
Interestingly, a semigroup with no element is an inverse semigroup (the necessary condition is vacuously satisfied). |
Interestingly, a semigroup with no element is an inverse semigroup (the necessary condition is vacuously satisfied). |
||
Revision as of 04:55, 31 May 2009
In mathematics, a semigroup with zero elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation.[1][2] However not all authors insist on the underlying set of a semigroup being non-empty.[3] One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from S × S to S. This operation vacuously satisfies the associativity property. The postulation of the existence of the empty semigroup gives more generality for certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup T is a subsemigroup of T becomes valid even when the intersection is empty.
Interestingly, a semigroup with no element is an inverse semigroup (the necessary condition is vacuously satisfied).
See also
Reference
- ^ A H Clifford, G B Preston (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. ISBN 978-0821802724
- ^ J M Howie (1976). An Introduction to Semigroup Theory. L.M.S.Monographs. Vol. 7. Academic Press. pp.2-3
- ^ P A Grillet (1995). Semigroups. CRC Press. ISBN 978-0824796624 pp.3-4