Abiordered set (otherwise known asboset) is amathematical object that occurs in the description of thestructure of the set ofidempotents in asemigroup.

The set of idempotents in a semigroup is a biordered set and every biordered set is the set of idempotents of some semigroup.^[1][2]A regular biordered set is a biordered set with an additional property. The set of idempotents in aregular semigroup is a regular biordered set, and every regular biordered set is the set of idempotents of some regular semigroup.^[1]

The concept and the terminology were developed byK S S Nambooripad in the early 1970s.^[3][4][1]In 2002, Patrick Jordan introduced the term boset as an abbreviation of biordered set.^[5] The defining properties of a biordered set are expressed in terms of twoquasiorders defined on the set and hence the name biordered set.

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."^[6] 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.^[7]

IfX andY aresets andρ ⊆X ×Y, letρ (y) = {x ∈X :xρy }.

LetE be aset in which apartialbinary operation, indicated by juxtaposition, is defined. IfD_E is thedomain of the partial binary operation onE thenD_E is arelation onE and (e,f) is inD_E if and only if the productef exists inE. The following relations can be defined inE:

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

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

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

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

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

IfT is anystatement aboutE involving the partial binary operation and the above relations inE, one can define the left-rightdual ofT denoted byT*. IfD_E issymmetric thenT* is meaningful wheneverT is.

The setE is called a biordered set if the followingaxioms and their duals hold for arbitrary elementse,f,g, etc. inE.

(B1)ω^r andω^l arereflexive andtransitive relations onE andD_E = (ω^r ∪ω^l) ∪ (ω^r ∪ω^l)⁻¹.

(B21) Iff is inω^r(e) thenf R fe ω e.

(B22) Ifgω^lf and iff andg are inω^r (e) thengeω^lfe.

(B31) Ifgω^rf andfω^re thengf = (ge)f.

(B32) Ifgω^lf and iff andg are inω^r (e) then (fg)e = (fe)(ge).

InM (e,f) =ω^l (e) ∩ ω^r (f) (theM-set ofe andf in that order), define a relation${\displaystyle \prec }$ by

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

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 thesandwich set ofe andf in that order.

(B4) Iff andg are inω^r (e) thenS(f,g)e =S (fe,ge).

We say that a biordered setE is anM-biordered set ifM (e,f) ≠ ∅ for alle andf inE. Also,E is called aregular biordered set ifS (e,f) ≠ ∅ for alle andf inE.

In 2012 Roman S. Gigoń gave a simple proof thatM-biordered sets arise fromE-inversive semigroups.^{[8][clarification needed]}

A subsetF of a biordered setE is a biordered subset (subboset) ofE ifF is a biordered set under the partial binary operation inherited fromE.

For anye inE the setsω^r (e),ω^l (e) andω (e) are biordered subsets ofE.^[1]

A mappingφ :E →F between two biordered setsE andF is a biordered set homomorphism (also called a bimorphism) if for all (e,f) inD_E we have (eφ) (fφ) = (ef)φ.

LetV be avector space and

E = {(A,B) |V =A ⊕B}

whereV =A ⊕B means thatA andB aresubspaces ofV andV is theinternal direct sum ofA andB. The partial binary operation ⋆ on E defined by

(A,B) ⋆ (C,D) = (A + (B ∩C), (B +C) ∩D)

makesE a biordered set. The quasiorders inE are characterised as follows:

(A,B)ω^r (C,D) ⇔A ⊇C

(A,B)ω^l (C,D) ⇔B ⊆D

The setE of idempotents in a semigroupS becomes a biordered set if a partial binary operation is defined inE as follows:ef is defined inE if and only ifef =e oref =f orfe =e orfe =f holds inS. IfS is a regular semigroup thenE is a regular biordered set.

As a concrete example, letS be the semigroup of all mappings ofX = { 1, 2, 3 } into itself. Let the symbol (abc) denote the map for which1 →a, 2 →b, and3 →c. The setE of idempotents inS 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 inE. AnX 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)				(122)
(133)	(111)	(222)	(333)	(122)	(133)			(133)		(133)
(121)	(111)	(222)	(333)	(121)		(121)	(323)			(121)
(323)	(111)	(222)	(333)			(121)	(323)		(323)	(323)
(113)	(111)	(222)	(333)		(113)			(113)	(223)	(113)
(223)	(111)	(222)	(333)				(223)	(113)	(223)	(223)
(123)	(111)	(222)	(333)	(122)	(133)	(121)	(323)	(113)	(223)	(123)

• Semigroup theory
• Algebraic structures
• Mathematical structures

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