Hemirings and Semirings: Definitions and Examples

Jonathan S. Golan
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Jonathan S. Golan: University of Haifa

Chapter 1 in Semirings and their Applications, 1999, pp 1-18 from Springer

Abstract: 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.

Keywords: Identity Element; Zero Divisor; Multiplicative Identity; Triangular Norm; Infinite Element (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-015-9333-5_1

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