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 A of a semigroup S is called an ideal of S if both SA and AS are subsets of A. Let I be an 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.[3] 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. In the Rees factor semigroup, the product of two elements in S \ I (the complement of S and I) is the same as their product in S if that product lies in S \ I; if otherwise, the product is given by the new element I.[4]
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 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. Cambridge Phil. Soc. 36: 387–400. 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), 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), "Extension of a semi-group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, 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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