Some Remarks About Semigroups of Contraction Mappings of a Finite Chain

On behalf of the Department of Mathematics and Statistics of the College of Arts and Sciences, you are cordially invited to a seminar to be conducted by Dr. Abdullahi Umar from Mathematics Department of The Petroleum Institute, Abu Dhabi, UAE.

Abstract:

The study of various (sub)-semigroups of transformations/mappings has made and is still making significant contributions to semigroup theory. The most notable classes are the THREE fundamental semigroups of transformations: the full symmetric semigroup, the partial symmetric semigroup and the symmetric inverse semigroup. The subsemigroups of order-preserving transformations, order-decreasing (extensive) transformations and their intersections are arguably the most studied classes. Others are the Baer-Levi and Croissot-Teissier semigroup. It is now established how counting certain natural equivalence classes in various semigroups of partial transformations of an n-set, leads to very interesting enumeration problems. Many numbers and triangle of numbers regarded as combinatorial gems like the Fibonacci number, Catalan number, Schröder number, Stirling numbers, Eulerian numbers, Narayana numbers, Lah numbers, etc., have all featured in these enumeration problems. These enumeration problems lead to many numbers and triangle of numbers in the Online encyclopedia of integer sequences (OEIS) but there are also others that are not yet or have just been recently recorded in OEIS. In this talk, we are going to focus on the combinatorial (enumerative) aspects of the classes of contraction transformation semigroups, which for some curious reason(s), until very recently, little is known about.

For further details, kindly contact [email protected].