Biordered set

If X and Y are 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:

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

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

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

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

${\displaystyle \omega =\omega ^{r}\,\cap \,\omega ^{l}}$

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.

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 ${\displaystyle \prec }$ by

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

Then the set

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

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 ).

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]

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.[1]

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.

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

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.