Discussiones Mathematicae General Algebra and Applications 24(2) (2004) 225-249
doi: 10.7151/dmgaa.1087

[BIBTex] [PDF] [PS]

GENERALIZED INFLATIONS AND NULL EXTENSIONS

Qiang Wang

School of Mathematics and Statistics
Carleton University
Ottawa, Ont., Canada K1S-5B6

e-mail: wang@math.carleton.ca

Shelly L. Wismath

Department of Mathematics and C.S.
University of Lethbridge
Lethbridge, Ab., Canada T1K-3M4

e-mail: wismaths@cs.uleth.ca

Abstract

An inflation of an algebra is formed by adding a set of new elements to each element in the original or base algebra, with the stipulation that in forming products each new element behaves exactly like the element in the base algebra to which it is attached. Clarke and Monzo have defined the generalized inflation of a semigroup, in which a set of new elements is again added to each base element, but where the new elements are allowed to act like different elements of the base, depending on the context in which they are used. Such generalized inflations of semigroups are closely related to both inflations and null extensions. Clarke and Monzo proved that for a semigroup base algebra which is a union of groups, any semigroup null extension must be a generalized inflation, so that the concepts of null extension and generalized inflation coincide in the case of unions of groups. As a consequence, the collection of all associative generalized inflations formed from algebras in a variety of unions of groups also forms a variety.

In this paper we define the concept of a generalized inflation for any type of algebra. In particular, we allow for generalized inflations of semigroups which are no longer semigroups themselves. After some general results about such generalized inflations, we characterize for several varieties of bands which null extensions of algebras in the variety are generalized inflations, and which of these are associative. These characterizations are used to produce examples which answer, in our more general setting, several of the open questions posed by Clarke and Monzo.

Keywords: inflation, generalized inflation, null extension, variety of semigroups, bands.

2000 Mathematics Subject Classification: 08A05, 20M07, 20M10.

References

[1] S. Bogdanović and S. Milić, Inflation of semigroups, Publ. Inst. Math. (N.S.) 41 (55) (1987), 63-73.
[2] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327-335.
[3] A. Christie, Q. Wang and S.L. Wismath, Class pperators involving inflations, preprint 2003.
[4] G.T. Clarke, Semigroup varieties of inflations of unions of groups, Semigroup Forum 23 (1981), 311-319.
[5] G.T. Clarke and R.A.R. Monzo, A generalization of the concept of an inflation of a semigroup, Semigroup Forum 60 (2000), 172-186.
[6] A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups, Vol. 1, Amer. Math. Soc., Providence, R.I, 1961.
[7] K. Denecke and S.L. Wismath, A characterization of k-normal varieties, Algebra Universalis 51 (2004), 395-409.
[8] E. Graczyńska, On normal and regular identities and hyperidentities, p. 107-135 in: ``Universal and Applied Algebra, (Turawa, 1988)'', World Scientific Publishing Co., Teaneck, NJ, 1989.
[9] I.I. Mel'nik, Nilpotent shifts of varieties, (in Russian), Mat. Zametki, 14 (1973), 703-712; English translation in: Math. Notes 14 (1973), 962-966.
[10] S. Milić, Inflation of algebras, p. 89-98 in: ``Procedings of the Conference Algebra and Logic, Sarajevo 1987'', Univ. Novi Sad, Novi Sad 1989.
[11] T. Tamura, Semigroups satisfying identity xy = f(x,y), Pacific J. Math. 31 (1969), 513-521.
[12] M. Yamada, A note on middle unitary semigroups, Kodai Math. Sem. Rep. 7 (1955), 49-52.

Received 17 May 2004
Revised 8 July 2004