Dominions and closed varieties of bands

Abstract

We show that all subvarieties of the variety of rectangular bands are closed in the variety of $n$-nilpotent extension of bands. Ahanger, Nabi and Shah in [1] have proved that the variety of regular bands is closed. In this paper, we improve this result and provide its simple and shorter proof. Finally, we show that all subvarieties of the variety of normal bands are closed in the variety of left [right] semiregular bands.

Keywords

• identity
• varieties
• zigzag equations
• dominion
• bands
• rectangular
• normal
• regular
• left semiregular

References

1. [1] S.A. Ahanger, M. Nabi and A.H. Shah, Closed and saturated varieties of semigroups, Comm. Algebra 51(1), 199-213, 2022.
2. [2] S.A. Ahanger and A.H. Shah, Epimorphisms, dominions and varietiess of bands, Semigroup Forum 100, 641-650, 2020.
3. [3] N. Alam and N.M. Khan, Special semigroup amalgams of quasi unitary subsemigroups and of quasi normal bands, Asian-Eur. J. Math. 6(1), (7 Pages), 2013.
4. [4] N. Alam and N.M. Khan, On closed and supersaturated semigroups, Comm. Algebra 42, 3137-3146, 2014.
5. [5] N. Alam and N.M. Khan, Epimorphism, closed and supersaturated semigroups, Malays. J. Math. Sci. 9(3), 409-416, 2015.
6. [6] A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups, Mathematical Surveys and Monographs 7(1) American Mathematical Society 1961, 1967.
7. [7] C. Fennemore, All varieties of bands, Semigroup Forum 1, 172-179, 1970.
8. [8] P.M. Higgins, Techniques of Semigroup Theory, Oxford University Press, Oxford, 1992.

9. [9] J.M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.
10. [10] J.R. Isbell, Epimorphisms and dominions, In: Proceedings of the conference on Categorical Algebra, La Jolla, 232-246, (1965), Lange and Springer, Berlin 1966.
11. [11] N.M. Khan,On saturated permutative varieties and consequences of permutation identities, J. Aust. Math. Soc.(Ser. A) 38, 186-197, 1985.
12. [12] M. Petrich, Lectures in Semigroups, Wiley, New York, 1977.
13. [13] H.E. Scheiblich, On epis and dominions of bands, Semigroup Forum 13, 103-114, 1976.