Biordered set

The concept and the terminology were developed by K S S Nambooripad in the early 1970s.{{cite book|last=Nambooripad|first=K S S|title=Structure of regular semigroups|publisher=University of Kerala, Thiruvananthapuram, India|date=1973|isbn=0-8218-2224-1}}{{cite journal|doi=10.1007/BF02194864|last=Nambooripad|first=K S S|date=1975|title=Structure of regular semigroups I . Fundamental regular semigroups|journal=Semigroup Forum|volume=9|issue=4|pages=354–363}}{{cite book|last=Nambooripad|first=K S S|title=Structure of regular semigroups – I|publisher=American Mathematical Society|date=1979|series=Memoirs of the American Mathematical Society|volume=224|isbn=978-0-8218-2224-1}}

In 2002, Patrick Jordan introduced the term boset as an abbreviation of biordered set.Patrick K. Jordan. On biordered sets, including an alternative approach to fundamental regular semigroups. Master's thesis, University of Sydney, 2002. The defining properties of a biordered set are expressed in terms of two quasiorders 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."{{cite book|last=Putcha|first=Mohan S|title=Linear algebraic monoids|publisher=Cambridge University Press|date=1988|series=London Mathematical Society Lecture Note Series|volume=133|pages=121–122|isbn=978-0-521-35809-5}} 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.{{cite journal|last=Easdown|first=David|date=1984|title=Biordered sets are biordered subsets of idempotents of semigroups|journal=Journal of the Australian Mathematical Society, Series A |volume=32|issue=2|pages=258–268|doi=10.1017/S1446788700022072 }}

If X and Y are sets and {{math|1=ρ ⊆ X × Y}}, let {{math|1=ρ ({{italics correction|y}}) = {{mset| 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 ({{italics correction|e,f}}) is in D_E if and only if the product ef exists in E. The following relations can be defined in E:

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

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

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

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

:$\backslash omega\; =\; \backslash omega^r\; \backslash ,\; \backslash cap\; \backslash ,\; \backslash omega^l$

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) {{mvar|ω^r}} and {{mvar|ω^l}} are reflexive and transitive relations on E and {{math|1=D_E = ({{italics correction|ω^r ∪ ω^l}}) ∪ ({{italics correction|ω^r ∪ ω^l}})⁻¹.}}

:(B21) If f is in ω^r({{italics correction|e}}) then f R fe ω e.

:(B22) If {{math|1=g ω^l f}} and if f and g are in {{math|1=ω^r ({{italics correction|e}})}} then {{math|1=ge ω^l fe}}.

:(B31) If {{math|1=g ω^r f}} and {{math|1=f ω^r e}} then gf = ({{italics correction|ge}})f.

:(B32) If {{math|1=g ω^l f}} and if f and g are in {{math|1=ω^r ({{italics correction|e}})}} then ({{italics correction|fg}})e = ({{italics correction|fe}})({{italics correction|ge}}).

In {{math|1=M ({{italics correction|e, f}}) = ω^l ({{italics correction|e}}) ∩ ω^r ({{italics correction|f}})}} (the M-set of e and f in that order), define a relation $\backslash prec$ by

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

Then the set

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

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

:(B4) If f and g are in ω^r ({{italics correction|e}}) then S({{italics correction|f, g}})e = S ({{italics correction|fe, ge}}).

We say that a biordered set E is an M-biordered set if M ({{italics correction|e, f}}) ≠ ∅ for all e and f in E.

Also, E is called a regular biordered set if S ({{italics correction|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.Gigoń, Roman (2012). "Some results on E-inversive semigroups". Quasigroups and Related Systems 20: 53-60.{{clarify|date=August 2014}}

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 {{math|1=ω^r ({{italics correction|e}}), ω^l ({{italics correction|e}})}} and {{math|1=ω ({{italics correction|e}})}} are biordered subsets of E.

A mapping φ : E → F between two biordered sets E and F is a biordered set homomorphism (also called a bimorphism) if for all ({{italics correction|e, f}}) in D_E we have ({{italics correction|eφ}}) ({{italics correction|fφ}}) = ({{italics correction|ef}})φ.

Let V be a vector space and

:{{math|1=E = {{mset|({{italics correction|A, B}}) |2= 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

:{{math|1=({{italics correction|A, B}}) ⋆ ({{italics correction|C, D}}) = ({{italics correction|A + ( B ∩ C}}), ({{italics correction|B + C}}) ∩ D )}}

makes E a biordered set. The quasiorders in E are characterised as follows:

:{{math|1=({{italics correction|A, B}}) ω^r ({{italics correction|C, D}}) ⇔ A ⊇ C}}

:{{math|1=({{italics correction|A, B}}) ω^l ({{italics correction|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 {{math|1=ef = e}} or {{math|1=ef = f}} or {{math|1=fe = e}} or {{math|1=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 {{math|1=X = {{mset| 1, 2, 3 }}}} into itself. Let the symbol (abc) denote the map for which {{math|1=1 → a, 2 → b,}} and {{math|1=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.

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Category:Semigroup theory

Category:Algebraic structures

Category:Mathematical structures