Rees factor semigroup

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]

Formal definition

A subset $I$ of a semigroup $S$ is called an ideal of $S$ if both $SI$ and $IS$ are subsets of $I$ . Let $I$ be an ideal of a semigroup $S$ . The relation $\rho$ in $S$ defined by

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

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

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

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

Example

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

Ideal extension

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]

References

↑ D. Rees (1940). "On semigroups". Proc. Camb. Phil. Soc. 36: 387&ndash, 400. 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), [https://books.google.com/books?id=2805q4tFiCkC&pg=PA60&dq=%22is+a+congruence%22 p. 60]
↑ 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", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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

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[1] D. Rees (1940). "On semigroups". Proc. Camb. Phil. Soc. 36: 387&ndash, 400. MR 2, 127

[2] 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.

[3] Lawson (1998), [https://books.google.com/books?id=2805q4tFiCkC&pg=PA60&dq=%22is+a+congruence%22 p. 60]

[4] Howie, John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9

[5] Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter (2002). The concise handbook of algebra. Springer. ISBN 978-0-7923-7072-7. (pp. 1–3)

[6] Gluskin, L.M. (2001) [1994], "Extension of a semi-group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4