Biordered set

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. David Easdown 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 D_E is the domain of the partial binary operation on E then D_E is a relation on E and (e,f) is in D_E if and only if the product ef exists in E. The following relations can be defined in E:

$\omega^r = \{(e,f) \, :\, fe = e\}$

$\omega^l = \{ (e,f)\, :\, ef = e \}$

$R = \omega^r\, \cap \, (\omega^r)^{-1}$

$L = \omega^l\, \cap \, (\omega^l)^{-1}$

$\omega = \omega^r \, \cap \, \omega^l$

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 D_E 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)  ω^r and ω^l are reflexive and transitive relations on E and D_E = ( ω^r ∪ ω ^l ) ∪ ( ω^r ∪ ω^l )⁻¹.

(B21)  If f is in ω^r( e ) then f R fe ω e.

(B22)  If g ω^l f and if f and g are in ω^r ( e ) then ge ω^l fe.

(B31)  If g ω^r f and f ω^r e then gf = ( ge )f.

(B32)  If g ω^l f and if f and g are in ω^r ( e ) then ( fg )e = ( fe )( ge ).

In M ( e, f ) = ω^l ( e ) ∩ ω^r ( f ) (the M-set of e and f in that order), define a relation $\prec$ by

$g \prec h\quad \Longleftrightarrow \quad eg \,\,\omega^r\,\, eh\,,\,\,\, gf\,\, \omega^l \,\,hf\,$.

Then the set

$S(e,f) = \{ h\in M(e,f) : g\prec h \text{ for all } g\in M(e,f) \}$

is called the sandwich set of e and f in that order.

(B4)  If f and g are in ω^r ( 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]

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 ω^r ( e ), ω^l ( 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 D_E 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 ) ω^r ( C, D ) ⇔ A ⊇ C

( A, B ) ω^l ( 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)	(122)	(121)	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	(233)	(113)	(223)	(223)
(123)	(111)	(222)	(333)	(122)	(133)	(121)	(323)	(113)	(223)	(123)

References

1. Nambooripad, K S S (1973). Structure of regular semigroups. University of Kerala, Thiruvananthapuram, India. ISBN 0-8218-2224-1.
2. Nambooripad, K S S (1975). "Structure of regular semigroups I . Fundamental regular semigroups". Semigroup Forum 9 (4): 354–363. doi:10.1007/BF02194864.
3. Nambooripad, K S S (1979). Structure of regular semigroups – I. Memoirs of the American Mathematical Society 224. American Mathematical Society. ISBN 978-0-8218-2224-1.
4. Patrick K. Jordan. On biordered sets, including an alternative approach to fundamental regular semigroups. Master’s thesis, University of Sydney, 2002.
5. Putcha, Mohan S (1988). Linear algebraic monoids. London Mathematical Society Lecture Note Series 133. Cambridge University Press. pp. 121–122. ISBN 978-0-521-35809-5.
6. Easdown, David (1984). "Biordered sets are biordered subsets of idempotents of semigroups". Journal of Australian Mathematical Society. Series A, 32 (2): 258–268.
7. Easdown, David (1985). "Biordered sets come from semigroups". Journal of Algebra 96: 581–91. doi:10.1016/0021-8693(85)90028-6.
8. Gigoń, Roman (2012). "Some results on E-inversive semigroups". Quasigroups and Related Systems 20: 53-60.