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 [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].[3] 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[4] 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. [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 (4): 387–400. doi:10.1017/S0305004100017436. 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.
- ↑ Lawson (1998) Inverse Semigroups: the theory of partial symmetries, page 60, World Scientific with Google Books link
- ↑ 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)
- ↑ Hazewinkel, Michiel, ed. (2001), "Extension of a semi-group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Extension_of_a_semi-group
- Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. ISBN 978-981-02-3316-7.
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