Rees factor semigroup (original) (raw)

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In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup.

Let S be a semigroup and I be an ideal of S. Using S and I one can construct a new semigroup by collapsing I into a single element while the elements of S outside of I retain their identity. The new semigroup obtained in this way is called the Rees factor semigroup of S modulo I and is denoted by S/I.

The concept of Rees factor semigroup was introduced by David Rees in 1940.[1][2]

A subset I {\displaystyle I} $I$ of a semigroup S {\displaystyle S} $S$ is called an ideal of S {\displaystyle S} $S$ if both S I {\displaystyle SI} $SI$ and I S {\displaystyle IS} $IS$ are subsets of I {\displaystyle I} $I$ (where S I = { s x ∣ s ∈ S and x ∈ I } {\displaystyle SI=\{sx\mid s\in S{\text{ and }}x\in I\}} $SI=\{sx\mid s\in S{\text{ and }}x\in I\}$ , and similarly for I S {\displaystyle IS} $IS$ ). Let I {\displaystyle I} $I$ be an ideal of a semigroup S {\displaystyle S} $S$ . The relation ρ {\displaystyle \rho } $\rho$ in S {\displaystyle S} $S$ defined by

x ρ y ⇔ either x = y or both x and y are in I

is an equivalence relation in S {\displaystyle S} $S$ . The equivalence classes under ρ {\displaystyle \rho } $\rho$ are the singleton sets { x } {\displaystyle \{x\}} $\{x\}$ with x {\displaystyle x} $x$ not in I {\displaystyle I} $I$ and the set I {\displaystyle I} $I$ . Since I {\displaystyle I} $I$ is an ideal of S {\displaystyle S} $S$ , the relation ρ {\displaystyle \rho } $\rho$ is a congruence on S {\displaystyle S} $S$ .[3] The quotient semigroup S / ρ {\displaystyle S/{\rho }} $S/{\rho }$ is, by definition, the Rees factor semigroup of S {\displaystyle S} $S$ modulo I {\displaystyle I} $I$ . For notational convenience the semigroup S / ρ {\displaystyle S/\rho } $S/\rho$ is also denoted as S / I {\displaystyle S/I} $S/I$ . The Rees factor semigroup[4] has underlying set ( S ∖ I ) ∪ { 0 } {\displaystyle (S\setminus I)\cup \{0\}} $(S\setminus I)\cup \{0\}$ , where 0 {\displaystyle 0} $0$ is a new element and the product (here denoted by ∗ {\displaystyle *} $*$ ) is defined by

s ∗ t = { s t if s , t , s t ∈ S ∖ I 0 otherwise . {\displaystyle s*t={\begin{cases}st&{\text{if }}s,t,st\in S\setminus I\\0&{\text{otherwise}}.\end{cases}}} $s*t={\begin{cases}st&{\text{if }}s,t,st\in S\setminus I\\0&{\text{otherwise}}.\end{cases}}$

The congruence ρ {\displaystyle \rho } $\rho$ on S {\displaystyle S} $S$ as defined above is called the Rees congruence on S {\displaystyle S} $S$ modulo I {\displaystyle I} $I$ .

Consider the semigroup S = { a, b, c, d, e } with the binary operation defined by the following Cayley table:

·	a	b	c	d	e
a	a	a	a	d	d
b	a	b	c	d	d
c	a	c	b	d	d
d	d	d	d	a	a
e	d	e	e	a	a

Let I = { a, d } which is a subset of S. Since

SI = { aa, ba, ca, da, ea, ad, bd, cd, dd, ed } = { a, d } ⊆ I

IS = { aa, da, ab, db, ac, dc, ad, dd, ae, de } = { a, d } ⊆ I

the set I is an ideal of S. The Rees factor semigroup of S modulo I is the set S/I = { b, c, e, I } with the binary operation defined by the following Cayley table:

·	b	c	e	I
b	b	c	I	I
c	c	b	I	I
e	e	e	I	I
I	I	I	I	I

A semigroup S is called an ideal extension of a semigroup A by a semigroup B if A is an ideal of S and the Rees factor semigroup S/A is isomorphic to B. [5]

Some of the cases that have been studied extensively include: ideal extensions of completely simple semigroups, of a group by a completely 0-simple semigroup, of a commutative semigroup with cancellation by a group with added zero. In general, the problem of describing all ideal extensions of a semigroup is still open.[6]

^ Rees, D. (1940). "On semigroups". Mathematical Proceedings of the Cambridge Philosophical Society. 36 (4): 387–400. doi:10.1017/S0305004100017436. S2CID 123038112. MR 2, 127
^ Clifford, Alfred Hoblitzelle; Preston, Gordon Bamford (1961). The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-0272-4. MR 0132791.
^ Lawson (1998) Inverse Semigroups: the theory of partial symmetries, page 60, World Scientific with Google Books link
^ Howie, John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9
^ Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter (2002). The concise handbook of algebra. Springer. ISBN 978-0-7923-7072-7.(pp. 1–3)
^ Gluskin, L.M. (2001) [1994], "Extension of a semi-group", Encyclopedia of Mathematics, EMS Press

Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. ISBN 978-981-02-3316-7.

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