ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications


Discussiones Mathematicae General Algebra and Applications 24(2) (2004) 225-249


Qiang Wang

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


Shelly L. Wismath

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



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.


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Received 17 May 2004
Revised 8 July 2004