Nambooripad order

In mathematics, Nambooripad order^[1] (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad^[2] in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig,^[3] some authors refer to it as Hartwig–Nambooripad order.^[4]

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 semigroups 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.^[5]

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 R_x be the Green R-class of S containing x. The relation R_x ≤ R_y 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

• a ≤ b if and only if R_a ≤ R_b and a = fb for some idempotent f in R_a

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.

Further generalization

Mitsch order

Mitsch generalized the definition of Nambooripad order to arbitrary semigroups.^{[6] [7]}

See also

• Regular semigroup
• Inverse semigroup

References

1. ^ Thomas Scott Blyth (2005). Lattices and ordered algebraic structures. Springer. pp. 228–232. ISBN 9781852339050. 
2. ^ K.S.S. Nambooripad (1980). "The natural partial order on a regular semigroup". Proceedings of the Edinburgh Mathematical Society 23: 249–260. 
3. ^ H. Mitsch (July 1986). "A natural partial order for semigroups". Proceedings of the American Mathematical Scociety 97 (3): 384–388. JSTOR 2046222. 
4. ^ J.B. Hickey (2004). "On regularity preservation on a semigroup". Bulletin of Australian Mathematical Society 69: 69–86. http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ69_01%2FS0004972700034274a.pdf&code=3112c0ca3657ba79a467b20ce538f628. Retrieved 11 April 2011. 
5. ^ H. Mitscjh (July 1986). "A natural partial order for semigroups". Proceedings of the American Mathematical Society 97 (3). http://www.ams.org/journals/proc/1986-097-03/S0002-9939-1986-0840614-0/S0002-9939-1986-0840614-0.pdf. Retrieved 11 April 2011. 
6. ^ Peter M. Higgins (1994). "The Mitsch order on a semigroup". Semigroup Forum 49 (1): 261–266. doi:10.1007/BF02573488. http://www.springerlink.com/content/v688p5un7m754108/. Retrieved 11 April 2011. 
7. ^ Mario Petrich (201). "Certain partial orders on semigroups". Czechoslovak Mathematical Journal 51 (2): 415–432. http://dml.cz/bitstream/handle/10338.dmlcz/127657/CzechMathJ_51-2001-2_15.pdf. Retrieved 11 April 2011.