Rees factor semigroup
Encyclopedia
In mathematics
, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor) 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.
is an equivalence relation in S. The equivalence classes under ρ are the singleton sets { x } with x not in I and the set I. Since I is an ideal of S, the relation ρ is a congruence on S. The quotient semigroup S/ρ is, by definition, the Rees factor semigroup of S modulo I. For notational convenience the semigroup S/ρ is also denoted as S/I.
The congruence ρ on S as defined above is called the Rees congruence on S modulo I.
Let I = { a, d } which is a subset of S. Since
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:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor) is a certain semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
constructed using a semigroup and an ideal of the semigroup.
Let S be a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
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
David Rees (mathematician)
David Rees ScD Cantab, FIMA, FRS is an emeritus professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter for many years....
in 1940.
Formal definition
A subset A of a semigroup S is called ideal of S if both SA and AS are subsets of A. Let I be ideal of a semigroup S. The relation ρ 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 ρ are the singleton sets { x } with x not in I and the set I. Since I is an ideal of S, the relation ρ is a congruence on S. The quotient semigroup S/ρ is, by definition, the Rees factor semigroup of S modulo I. For notational convenience the semigroup S/ρ is also denoted as S/I.
The congruence ρ 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 Calyley 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 | d | I |
| c | c | b | d | I |
| e | e | e | a | 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.