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.

Formal definition

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

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

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

[math]\displaystyle{ s * t = \begin{cases} st & \text{if } s, t, st \in S \setminus I \\ 0 & \text{otherwise}. \end{cases} }[/math]

The congruence [math]\displaystyle{ \rho }[/math] on [math]\displaystyle{ S }[/math] as defined above is called the Rees congruence on [math]\displaystyle{ S }[/math] modulo [math]\displaystyle{ I }[/math].

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.

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.

References

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