Idempotent Structures Of Semigroup Of Singular - Regular Matrices

Adenike Olusola Adeniji

Adenike Olusola Adeniji University of Abuja, Abuja

Keywords: Tridempotents, Idempotents, Normal Semigroup, Band, Principal Diagonal

Abstract

Let S be a semigroup of singular matrices over a subset of the set of integers. The idempotent
elements of S are classified into tridempotent, fractional idempotent and skew idempotent
elements. Matrices are generally known to be non-commutative but an example of commutative
matrices is established in this work. Matrix multiplication and its axioms are employed
to establish the results. Regularity was first established and the condition for regularity of
each idempotent structure obtained is discussed. Some of the structures obtained are used to
establish bands of regular elements. The fractional component of S is obtained combinatorially
by choosing a number with its sign for every row of each matrix. There are m values of
matrices with first and second rows being equal, which are removed from the set since their
multiplication gives zero. The satisfaction of the condition for commutativity implies that all
the matrices have the same characteristic equation and all are of a specified order n. This is
evident in the fact that the product of the principal diagonal, D1 of matrix A is the same as
the product of the principal diagonal D2 of matrix B. Also, the product of the off- diagonal of
matrices A and B are the same.

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References

[1] A. H. Clifford,Bands of semigroups, Proc. Amer. Math. Soc., 5(1954) 499-504.
[2] A.R. Richardson, Conjugate Matrices, The Quarterly Journal of Mathematics, os-7(1)(1936) 256-270,
https://doi.org/10.1093/qmath/os-7.1.256
[3] G. Garba, Idempotents in partial transformation semigroups, Proceedings of the Royal Society of Edinburgh:
Section A Mathematics, 116(3-4), (1990)359-366.doi:10.1017/S0308210500031553.
[4] J.M. Howie & R. McFadden, Idempotent rank in finite full transformation semigroups, Proceedings
of the Royal Society of Edinburgh: Section A Mathematics, 114(3-4)(1990) 161-167.
doi:10.1017/S0308210500024355.
[5] J.M. Howie, Fundamentals of semigroups theory, LMS Monographs, New Series, No. 12, Clarendon
Press, Oxford, 1995.
[6] M. V. Lawson, Inverse semigroups, the theory of partial symmetries, World Scientific, 1998.
[7] O. Ganyushkin and V. Mazorchuk, Introduction to classical finite transformation semigroups, Springer
- Verlag London Limited, (2009).
[8] P.M. Higgins, Idempotent depth in semigroups of order-preserving mappings, Proceedings of The Royal
Society A: Mathematical, Physical and Engineering Sciences, 124(1994)1045-1058.
[9] R. Kehinde & A. D. Adeshola, The order of the set of idempotent elements of semigroup of partial
isometries of a finite chain, International Journal of Science and Research, 2(5)(2013) 291 - 292.