Rees factor semigroup explained

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

I of a semigroup S is called an ideal of S if both SI and IS are subsets of I (where SI = \{sx \mid s \in S and x \in I\} , and similarly for IS ). 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 moduloI . For notational convenience the semigroup S/\rho is also denoted as S/I . The Rees factorsemigroup has underlying set (S \setminus I) \cup \{ 0 \} , where 0 is a new element and the product (here denoted by * ) is defined bys * t = \begin{cases} st & if s, t, st \in S \setminus I \\ 0 & 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 = 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 = which is a subset of S. Since

SI = = ⊆ I

IS = = ⊆ I

the set I is an ideal of S. The Rees factor semigroup of S modulo I is the set S/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. ^[4]

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

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

Notes and References

1. D. Rees. 1940. On semigroups. Proc. Camb. Phil. Soc.. 36. 4 . 387 - 400 . 10.1017/S0305004100017436 . 123038112 . MR 2, 127
2. Book: Clifford . Alfred Hoblitzelle . Alfred H. Clifford . Preston . Gordon Bamford . Gordon Preston . The algebraic theory of semigroups. Vol. I . . Providence, R.I. . Mathematical Surveys, No. 7 . 978-0-8218-0272-4 . 0132791 . 1961 .
3. Lawson (1998) Inverse Semigroups: the theory of partial symmetries, page 60, World Scientific with Google Books link
4. Book: Mikhalev . Aleksandr Vasilʹevich. Pilz . Günter. The concise handbook of algebra. . 978-0-7923-7072-7 . 2002 . (pp. 1 - 3)