Generalized inflations and null extensions

Qiang Wang; Shelly L. Wismath

Discussiones Mathematicae - General Algebra and Applications (2004)

  • Volume: 24, Issue: 2, page 225-249
  • ISSN: 1509-9415

Abstract

top
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.

How to cite

top

Qiang Wang, and Shelly L. Wismath. "Generalized inflations and null extensions." Discussiones Mathematicae - General Algebra and Applications 24.2 (2004): 225-249. <http://eudml.org/doc/287739>.

@article{QiangWang2004,
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.},
author = {Qiang Wang, Shelly L. Wismath},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {inflation; generalized inflation; null extension; variety of semigroups; bands; extensions of universal algebras; inflations of universal algebras; inflations of semigroups},
language = {eng},
number = {2},
pages = {225-249},
title = {Generalized inflations and null extensions},
url = {http://eudml.org/doc/287739},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Qiang Wang
AU - Shelly L. Wismath
TI - Generalized inflations and null extensions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2004
VL - 24
IS - 2
SP - 225
EP - 249
AB - 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.
LA - eng
KW - inflation; generalized inflation; null extension; variety of semigroups; bands; extensions of universal algebras; inflations of universal algebras; inflations of semigroups
UR - http://eudml.org/doc/287739
ER -

References

top
  1. [1] S. Bogdanović and S. Milić, Inflation of semigroups, Publ. Inst. Math. (N.S.) 41 (55) (1987), 63-73. Zbl0637.20032
  2. [2] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327-335. Zbl0842.08007
  3. [3] A. Christie, Q. Wang and S.L. Wismath, Class pperators involving inflations, preprint 2003. 
  4. [4] G.T. Clarke, Semigroup varieties of inflations of unions of groups, Semigroup Forum 23 (1981), 311-319. Zbl0486.20033
  5. [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. Zbl0951.20045
  6. [6] A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups, Vol. 1, Amer. Math. Soc., Providence, R.I, 1961. Zbl0111.03403
  7. [7] K. Denecke and S.L. Wismath, A characterization of k-normal varieties, Algebra Universalis 51 (2004), 395-409. Zbl1080.08002
  8. [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. Zbl0747.08006
  9. [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. [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. [11] T. Tamura, Semigroups satisfying identity xy = f(x,y), Pacific J. Math. 31 (1969), 513-521. Zbl0186.31102
  12. [12] M. Yamada, A note on middle unitary semigroups, Kodai Math. Sem. Rep. 7 (1955), 49-52. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.