Nambooripad order
Encyclopedia
In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup
discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order.
Nambooripad's partial order is a generalisation of an earlier known partial order on the set of idempotents in any semigroup
. The partial order on the set E of idempotents in a semigroup S is defined as follows: For any e and f in E, e ≤ f if and
only if e = ef = fe. Vagner in 1952 had extended this to inverse semigroup
s as follows: For any a and b in an inverse semigroup S, a ≤ b if and only if a = eb for some idempotent e in S. This partial order is compatible with multiplication on both sides, that is, if a ≤ b then ac ≤ bc and ca ≤ cb for all c in S. Nambooripad extended these definitions to regular semigroups. In general Nambooripad's order in a regular semigroup is not compatible with multiplication. It is compatible with multiplication only if the semigroup is pseudo-inverse.
of S containing x.
The relation Rx ≤ Ry defined by xS1 ⊆ yS1 is a partial order in the collection of Green R-classes
in S. For a and b in S the relation ≤ defined by
is a partial order in S. This is a natural partial order in S.
For a and b in S the relation ≤ defined by
is a partial order in S. This is a natural partial order in S.
is a partial order in S. This is a natural partial order in S.
Regular semigroup
A regular semigroup is a semigroup S in which every element is regular, i.e., for each element a, there exists an element x such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.- Origins...
discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order.
Nambooripad's partial order is a generalisation of an earlier known partial order on the set of idempotents in any semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
. The partial order on the set E of idempotents in a semigroup S is defined as follows: For any e and f in E, e ≤ f if and
only if e = ef = fe. Vagner in 1952 had extended this to inverse semigroup
Inverse semigroup
In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
s as follows: For any a and b in an inverse semigroup S, a ≤ b if and only if a = eb for some idempotent e in S. This partial order is compatible with multiplication on both sides, that is, if a ≤ b then ac ≤ bc and ca ≤ cb for all c in S. Nambooripad extended these definitions to regular semigroups. In general Nambooripad's order in a regular semigroup is not compatible with multiplication. It is compatible with multiplication only if the semigroup is pseudo-inverse.
Definitions
The partial order in a regular semigroup discovered by Nambooripad can be defined in several equivalent ways. Three of these definitions are given below. The equivalence of these definitions and other definitions have been established by Mitch.Definition (Nambooripad)
Let S be any regular semigroup and S1 be the semigroup obtained by adjoining the identity 1 to S. For any x in S let Rx be the Green R-classGreen's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
of S containing x.
The relation Rx ≤ Ry defined by xS1 ⊆ yS1 is a partial order in the collection of Green R-classes
Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
in S. For a and b in S the relation ≤ defined by
- a ≤ b if and only if Ra ≤ Rb and a = fb for some idempotent f in Ra
is a partial order in S. This is a natural partial order in S.
Definition (Hartwig)
For any element a in a regular semigroup S, let V(a) be the set of inverses of a, that is, the set of all x in S such that axa = a and xax = x.For a and b in S the relation ≤ defined by
- a ≤ b if and only if a'a = a'b and aa' = ba' for some a' in V(a)
is a partial order in S. This is a natural partial order in S.
Definition (Mitsch)
For a and b in in a regular semigroup S the relation ≤ defined by- a ≤ b if and only if a = xa = xb = by for some element x and y in S
is a partial order in S. This is a natural partial order in S.