Abstract
A semigroup (M, *) consists of a nonempty set M on which an associative operation * is defined. If M is a semigroup in which there exists an element e satisfying m * e = m = e * m for all m ∈ M, then M is called a monoid having identity element e. This element can easily seen to be unique, and is usually denoted by 1m- Note that a semigroup (M, *) which is not a monoid can be canonically embedded in a monoid M′ - M ∪ {e} where e is some element not in M, and where the operation * is extended to an operation on M′ by defining e * M′ = M′ = M′ * e for all m′ ∈ m′. An element m of M idempotent if and only if m * m = m. A semigroup (M, *) is commutative if and only if m * M′ = M′ * m for all m.m′ ∈ M.
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© 1999 Springer Science+Business Media Dordrecht
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Golan, J.S. (1999). Hemirings and Semirings: Definitions and Examples. In: Semirings and their Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9333-5_1
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DOI: https://doi.org/10.1007/978-94-015-9333-5_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5252-0
Online ISBN: 978-94-015-9333-5
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