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Hemirings and Semirings: Definitions and Examples

In: Semirings and their Applications

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  • Jonathan S. Golan

    (University of Haifa)

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.

Suggested Citation

  • Jonathan S. Golan, 1999. "Hemirings and Semirings: Definitions and Examples," Springer Books, in: Semirings and their Applications, chapter 1, pages 1-18, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9333-5_1
    DOI: 10.1007/978-94-015-9333-5_1
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