Idempotents in Semigroups: Structure, Classification, Extension and Applications
Main Article Content
Abstract
This paper examines the structure, classification and extension of idempotents in semigroups, exploring their fundamental algebraic structures. Different identification and extension of idempotents in semigroups are presented. The various forms of examining idempotent behavior under different semigroup operations and their relationships with other elements are reviewed. Also, the distribution of idempotents within different classes of semigroups, such as regular semigroups, finite semigroups, and infinite semigroups are explained. The findings provides a deeper understanding of idempotent in semigroup structure, giving insights into the role of idempotents in more complex algebraic systems. Applications of idempotents in various fields are also presented.
Article Details
References
Abubakar, R. B. (2022). Congruence and Kernels of left restriction semigroups in ℘????????????−???????? .THE
THE COLLOQUIUM, 10(1), 266- 301 https://www.ajol.info/index.php/colloq
Abubakar, R.B., & David, E. E. (2016). 0-simple semigroup in graph theory. Akoka journal of Pure and Applied Science Education , 14(1) ,12-24
Almeida, J., & Zeitoun, M. (2010). The pseudovariety J is hyperdecidable .Theoretical Informatics and Applications, 44(3), 439-458.
Arkowitz, M. (1995). Co-H-Spaces. Handbook of Algebraic Topology , 1145-1173
Auinger, K., & Steinberg, B. (2011). A new approach to the Karoubi envelope and applications. Pacific Journal of Mathematics, 256(1), 1-20.
Barto, L., & Kozik, M. (2014). Constraint satisfaction problems solvable by local consistency methods. Journal of the ACM (JACM), 61(1), 1-19.
Blyth, T.S., & Almeida, S. M.H. (2002). Regular semigroups with skew pairs of idempotents. Semigroup Forum, 264 - 274 https://doi.org/10.1007/s002330010112
Bui-Xuan, B., & Dejean, A. (2012). The idempotent rank in the endomorphism semigroup of a uniform partition. Semigroup Forum, 85(1), 87-103.
Bulman-Fleming, S., & Jones, P. R. (2013). Idempotent-generated completely 0-simple semigroups. Journal of Algebra, 379, 57-69.
Carlson, J. F. (1996). Idempotent modules In: Modules and Algebras. Lectures in Mathematics ETHZurich Birkhauser Basel https://doi.org/10.1007/978-3-0348-9189-9_12
Chrislock, J. L.(1969).On medial semigroups. Journal of Algebra 12, 1-9 https://doi.org/10.1016/0021-8693(69)90013-1
Costa, A., & Nogueira, C. (2017). On the ranks of certain semigroups of transformations preserving a uniform partition. Communications in Algebra, 45(12), 5112-5124.
Dolinka, I. (2015). A note on endomorphisms and idempotent-generated subsemigroups. Semigroup Forum, 91(2), 375-379.
Fleischer, L., & Schwartz, R. (2013). On the structure of semigroups generated by idempotents. Semigroup Forum, 87(3), 615-625.
Gan, H., & Zhang, W. T. (2014). On certain idempotent-generated semigroups. Semigroup Forum, 88(1), 157-167.
Gomes, G. M. S., & Gould, V. (2011). Idempotent factorizations in some classes of regular semigroups. Semigroup Forum, 82(1), 8-28.
Harju, T., & Peter, I. (2012). Transformation semigroups with restricted rank. Journal of Pure and Applied Algebra, 216(2), 353-365.
Higgins, P. M. (2015). Techniques of semigroup theory. Clarendon Press.
Hollings, C. & Gould, V. (2011). Restriction semigroups and inductive constellations. Comm Algebra ,38 , 261-287
Howie, J. (2013). Fundamentals of semigroup theory. Oxford university press
Howie, J. M., & Marques-Smith, L. (2011). Idempotent rank in the endomorphism semigroup of a non-uniform partition. Semigroup Forum, 82(2), 288-300.
Kolokoltsov,V.N. &Maslov, V.P.(1997). Idempotent analysis and its applications. Klover Academic Publishers
Lallement, G. (2018). Semigroups and combinatorial applications. Wiley.
Lawson, J.D. (2004). Idempotent analysis and continuous semilattices. Theoretical Computer Science , 316, 75-87
Lawson, M. V. (2021). Inverse semigroups: The theory of partial symmetries. World Scientific.
Linder, C. C. (1991).Embedding theorems for partial Latin square. Annals of Discrete Maths, 46, 217-265
McEneaney,W.M. (2009). Idempotent algorithm for discrete-time stochastic control through distributed dynamic programming .Proceeding of the 48th IEEC conference on Decision and control (CDC) held jointly with 2009 28th Chinese control conference
McFadden, C., & Ruškuc, N. (2015). Idempotent-generated semigroups of order-preserving transformations. 32(1), 121-136.
Pin, J. E. (2012). Mathematical foundations of automata theory. Elsevier.
Ren, Q., & Xu, Y. (2020). Idempotent presentation in transformation semigroups. Journal of Algebra, 556, 362-378.
Szendrei, M. B., & Zhang, H. (2016). Congruences on finite idempotent semigroups. Journal of Pure and Applied Algebra, 220(9), 3420-3432.
Thomas, J., Chandrasekaran,V. M., & Kandaiyan, I. (2018). A study on regular semigroups and its idempotents. Int. Journal of Engineering and Technology https://doi:10.14419/ijet.v7i4.10.21214
Yamada, Y. (2023). New results on the ranks of certain semigroups of transformations. Semigroup Forum, 107(1), 1-15.