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In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so all-pervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'". The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility.
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In mathematics, Nambooripad order is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order. "Natural" here means that the order is defined in terms of the operation on the semigroup.
In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all x in a semigroup S, there exists a positive integer n and a subgroup G of S such that xn belongs to G.
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In abstract algebra, an E-dense semigroup is a semigroup in which every element a has at least one weak inversex, meaning that xax = x. The notion of weak inverse is weaker than the notion of inverse used in a regular semigroup.
In the area of mathematics known as semigroup theory, an E-semigroup is a semigroup in which the idempotents form a subsemigroup.
In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigroups and varieties of finite ordered monoids are defined similarly.
References
- ↑ J. Almeida, J.-É. Pin and P. Weil Semigroups whose idempotents form a subsemigroup updated version of Almeida, J.; Pin, J.-É.; Weil, P. (2008). "Semigroups whose idempotents form a subsemigroup". Mathematical Proceedings of the Cambridge Philosophical Society. 111 (2): 241. doi:10.1017/S0305004100075332. S2CID 6344747.
- ↑ Hall, T. E. (1969). "On regular semigroups whose idempotents form a subsemigroup". Bulletin of the Australian Mathematical Society. 1 (2): 195–208. doi: 10.1017/s0004972700041447 .
- ↑ Clifford, A. H.; Hofmann, K. H.; Mislove, M. W., eds. (1996). Semigroup Theory and Its Applications: Proceedings of the 1994 Conference Commemorating the Work of Alfred H. Clifford . Cambridge University Press. p. 70. ISBN 9780521576697.
- ↑ P.H.H. Fantham (1960). "On the Classification of a Certain Type of Semigroup". Proceedings of the London Mathematical Society. 1: 409–427. doi:10.1112/plms/s3-10.1.409.
- 1 2 J.M. Howie (1976). An introduction to semigroup theory. London: Academic Press. pp. 186–211.
- 1 2 P.A. Grillet. Semigroups: An introduction to structure theory. New York: Marcel Dekker, Inc. p. 341.