Fundamentals of Semigroup Theory

Howie, J. M. (1995). Fundamentals of Semigroup Theory. Oxford University Press.

Type

Book

Intellectual base

Textbook

Year

1995

DOI

10.1093/oso/9780198511946.001.0001

ISBN

9780198511946

Link

https://archive.org/details/fundamentalsofse0000howi

Cited from

mathematics

Cited by

5 citations across 5 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Identity Element
  • Neither implies the other, and neither is unique: the operation x • y = y makes every member a left identity while admitting no right identity at all on a carrier with more than one element.

    This sourceTreats left and right identities as logically independent conditions, gives the left-meets-right uniqueness argument, the right-zero semigroup in which every element is a left identity and none a right identity, the adjunction of an identity to a semigroup, and the monoid-homomorphism obligation to preserve the unit.

• Monoid
  • If e and e' are both two-sided identities for the same operation, then e = e • e' = e', so a monoid has exactly one identity; "the" identity is precise language, not a convenient abuse of it.

    This sourceLondon Mathematical Society Monographs, New Series 12. Clarendon Press, Oxford, 1995. Supplies the uniqueness of a two-sided identity, the adjunction of an identity to any semigroup, the monoid homomorphism conditions including preservation of the unit, and the Cayley embedding into the endofunction monoid by translation.

Domain-specific

• Absorbing element
  • Left, right and two-sided absorbing elements differ in noncommutative systems, and additive zero is usually an identity rather than absorbing.
• Bicyclic semigroup
  • The same object is realized by partial shifts on the nonnegative integers and as a syntactic monoid associated with balanced-parenthesis structure.
• Catholic Semigroup
  • It is not orthodoxy, which asks that the idempotents of a regular semigroup form a subsemigroup.

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