Biordered set
This article's tone or style may not reflect the encyclopedic tone used on Wikipedia. (November 2012) |
A biordered set ("boset") is a mathematical object that occurs in the description of the structure of the set of idempotents in a semigroup. The concept and the terminology were developed by K S S Nambooripad in the early 1970s.[1][2][3] The defining properties of a biordered set are expressed in terms of two quasiorders defined on the set and hence the name biordered set. Patrick Jordan, while a master's student at University of Sydney, introduced in 2002 the term boset as an abbreviation of biordered set.[4]
According to Mohan S. Putcha, "The axioms defining a biordered set are quite complicated. However, considering the general nature of semigroups, it is rather surprising that such a finite axiomatization is even possible."[5] Since the publication of the original definition of the biordered set by Nambooripad, several variations in the definition have been proposed. simplified the definition and formulated the axioms in a special arrow notation invented by him.[6]
The set of idempotents in a semigroup is a biordered set and every biordered set is the set of idempotents of some semigroup.[3][7] A regular biordered set is a biordered set with an additional property. The set of idempotents in a regular semigroup is a regular biordered set, and every regular biordered set is the set of idempotents of some regular semigroup.[3]
Definition[]
The formal definition of a biordered set given by Nambooripad[3] requires some preliminaries.
Preliminaries[]
If X and Y be sets and ρ⊆ X × Y, let ρ ( y ) = { x ∈ X : x ρ y }.
Let E be a set in which a partial binary operation, indicated by juxtaposition, is defined. If DE is the domain of the partial binary operation on E then DE is a relation on E and (e,f) is in DE if and only if the product ef exists in E. The following relations can be defined in E:
If T is any statement about E involving the partial binary operation and the above relations in E, one can define the left-right dual of T denoted by T*. If DE is symmetric then T* is meaningful whenever T is.
Formal definition[]
The set E is called a biordered set if the following axioms and their duals hold for arbitrary elements e, f, g, etc. in E.
- (B1) ωr and ωl are reflexive and transitive relations on E and DE = ( ωr ∪ ω l ) ∪ ( ωr ∪ ωl )−1.
- (B21) If f is in ωr( e ) then f R fe ω e.
- (B22) If g ωl f and if f and g are in ωr ( e ) then ge ωl fe.
- (B31) If g ωr f and f ωr e then gf = ( ge )f.
- (B32) If g ωl f and if f and g are in ωr ( e ) then ( fg )e = ( fe )( ge ).
In M ( e, f ) = ωl ( e ) ∩ ωr ( f ) (the M-set of e and f in that order), define a relation by
- .
Then the set
is called the sandwich set of e and f in that order.
- (B4) If f and g are in ωr ( e ) then S( f, g )e = S ( fe, ge ).
M-biordered sets and regular biordered sets[]
We say that a biordered set E is an M-biordered set if M ( e, f ) ≠ ∅ for all e and f in E. Also, E is called a regular biordered set if S ( e, f ) ≠ ∅ for all e and f in E.
In 2012 Roman S. Gigoń gave a simple proof that M-biordered sets arise from E-inversive semigroups.[8][clarification needed]
Subobjects and morphisms[]
Biordered subsets[]
A subset F of a biordered set E is a biordered subset (subboset) of E if F is a biordered set under the partial binary operation inherited from E.
For any e in E the sets ωr ( e ), ωl ( e ) and ω ( e ) are biordered subsets of E.[3]
Bimorphisms[]
A mapping φ : E → F between two biordered sets E and F is a biordered set homomorphism (also called a bimorphism) if for all ( e, f ) in DE we have ( eφ ) ( fφ ) = ( ef )φ.
Illustrative examples[]
Vector space example[]
Let V be a vector space and
- E = { ( A, B ) | V = A ⊕ B }
where V = A ⊕ B means that A and B are subspaces of V and V is the internal direct sum of A and B. The partial binary operation ⋆ on E defined by
- ( A, B ) ⋆ ( C, D ) = ( A + ( B ∩ C ), ( B + C ) ∩ D )
makes E a biordered set. The quasiorders in E are characterised as follows:
- ( A, B ) ωr ( C, D ) ⇔ A ⊇ C
- ( A, B ) ωl ( C, D ) ⇔ B ⊆ D
Biordered set of a semigroup[]
The set E of idempotents in a semigroup S becomes a biordered set if a partial binary operation is defined in E as follows: ef is defined in E if and only if ef = e or ef= f or fe = e or fe = f holds in S. If S is a regular semigroup then E is a regular biordered set.
As a concrete example, let S be the semigroup of all mappings of X = { 1, 2, 3 } into itself. Let the symbol (abc) denote the map for which 1 → a, 2 → b, and 3 → c. The set E of idempotents in S contains the following elements:
- (111), (222), (333) (constant maps)
- (122), (133), (121), (323), (113), (223)
- (123) (identity map)
The following table (taking composition of mappings in the diagram order) describes the partial binary operation in E. An X in a cell indicates that the corresponding multiplication is not defined.
∗ | (111) | (222) | (333) | (122) | (133) | (121) | (323) | (113) | (223) | (123) |
---|---|---|---|---|---|---|---|---|---|---|
(111) | (111) | (222) | (333) | (111) | (111) | (111) | (333) | (111) | (222) | (111) |
(222) | (111) | (222) | (333) | (222) | (333) | (222) | (222) | (111) | (222) | (222) |
(333) | (111) | (222) | (333) | (222) | (333) | (111) | (333) | (333) | (333) | (333) |
(122) | (111) | (222) | (333) | (122) | (133) | (122) | X | X | X | (122) |
(133) | (111) | (222) | (333) | (122) | (133) | X | X | (133) | X | (133) |
(121) | (111) | (222) | (333) | (121) | X | (121) | (323) | X | X | (121) |
(323) | (111) | (222) | (333) | X | X | (121) | (323) | X | (323) | (323) |
(113) | (111) | (222) | (333) | X | (113) | X | X | (113) | (223) | (113) |
(223) | (111) | (222) | (333) | X | X | X | (223) | (113) | (223) | (223) |
(123) | (111) | (222) | (333) | (122) | (133) | (121) | (323) | (113) | (223) | (123) |
References[]
- ^ Nambooripad, K S S (1973). Structure of regular semigroups. University of Kerala, Thiruvananthapuram, India. ISBN 0-8218-2224-1.
- ^ Nambooripad, K S S (1975). "Structure of regular semigroups I . Fundamental regular semigroups". Semigroup Forum. 9 (4): 354–363. doi:10.1007/BF02194864.
- ^ a b c d e Nambooripad, K S S (1979). Structure of regular semigroups – I. Memoirs of the American Mathematical Society. Vol. 224. American Mathematical Society. ISBN 978-0-8218-2224-1.
- ^ Patrick K. Jordan. On biordered sets, including an alternative approach to fundamental regular semigroups. Master's thesis, University of Sydney, 2002.
- ^ Putcha, Mohan S (1988). Linear algebraic monoids. London Mathematical Society Lecture Note Series. Vol. 133. Cambridge University Press. pp. 121–122. ISBN 978-0-521-35809-5.
- ^ Easdown, David (1984). "Biordered sets are biordered subsets of idempotents of semigroups". Journal of the Australian Mathematical Society, Series A. 32 (2): 258–268.
- ^ Easdown, David (1985). "Biordered sets come from semigroups". Journal of Algebra. 96 (2): 581–91. doi:10.1016/0021-8693(85)90028-6.
- ^ Gigoń, Roman (2012). "Some results on E-inversive semigroups". Quasigroups and Related Systems 20: 53-60.
- Semigroup theory
- Algebraic structures
- Mathematical structures