Geodesic
Generalized inflations and null extensions
Discussiones Mathematicae. General Algebra and Applications, Tome 24 (2004) no. 2, pp. 225-249

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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.
Keywords: inflation, generalized inflation, null extension, variety of semigroups, bands 
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Wang, Qiang; Wismath, Shelly. Generalized inflations and null extensions. Discussiones Mathematicae. General Algebra and Applications, Tome 24 (2004) no. 2, pp. 225-249. http://geodesic.mathdoc.fr/item/DMGAA_2004_24_2_a5/