Rees factor semigroup

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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 I of a semigroup S is called an ideal of S if both SI and IS are subsets of I (where SI={sx∣s∈S and x∈I}, and similarly for IS). 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. The Rees factor semigroup[4] has underlying set (S∖I)∪{0}, where 0 is a new element and the product (here denoted by *) is defined by s*t={stif s,t,st∈S∖I0otherwise. 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

  1. ↑ D. Rees (1940). "On semigroups". Proc. Camb. Phil. Soc. 36 (4): 387–400. doi:10.1017/S0305004100017436. S2CID 123038112. MR 2, 127
  2. ↑ 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.
  3. ↑ Lawson (1998) Inverse Semigroups: the theory of partial symmetries, page 60, World Scientific with Google Books link
  4. ↑ Howie, John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9
  5. ↑ Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter (2002). The concise handbook of algebra. Springer. ISBN 978-0-7923-7072-7.(pp. 1–3)
  6. ↑ Gluskin, L.M. (2001) [1994], "Extension of a semi-group", Encyclopedia of Mathematics, EMS Press
  • Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. ISBN 978-981-02-3316-7.

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