Cancellative orders

D Easdown; Victoria Gould

Cancellative orders

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every q is an element of Q can be written as q = a*b for some a, b is an element of S where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the D-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S.

Original language	English
Pages (from-to)	185-195
Number of pages	11
Journal	Semigroup forum
Volume	55
Issue number	2
Publication status	Published - 1997

Keywords

SEMIGROUPS
QUOTIENTS

Cite this

@article{dcea5ec752414fef86007d39d2bd5928,

title = "Cancellative orders",

abstract = "A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every q is an element of Q can be written as q = a*b for some a, b is an element of S where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the D-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S.",

keywords = "SEMIGROUPS, QUOTIENTS",

author = "D Easdown and Victoria Gould",

year = "1997",

language = "English",

volume = "55",

pages = "185--195",

journal = "Semigroup forum",

issn = "0037-1912",

publisher = "Springer",

number = "2",

}

TY - JOUR

T1 - Cancellative orders

AU - Easdown, D

AU - Gould, Victoria

PY - 1997

Y1 - 1997

N2 - A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every q is an element of Q can be written as q = a*b for some a, b is an element of S where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the D-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S.

AB - A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every q is an element of Q can be written as q = a*b for some a, b is an element of S where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the D-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S.

KW - SEMIGROUPS

KW - QUOTIENTS

M3 - Article

SN - 0037-1912

VL - 55

SP - 185

EP - 195

JO - Semigroup forum

JF - Semigroup forum

IS - 2

ER -