Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

Auinger, Karl

Glasgow mathematical journal, Tome 35 (1993) no. 1, pp. 25-37

Résumé

For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

Détail

DOI : 10.1017/S0017089500009538

Bibliographie

[1] 1.Adair, C. L., Bands with involution, J. Algebra 75 (1982), 297–314. Google Scholar | DOI

[2] 2.Auinger, K., Free strict inverse semigroups, J. Algebra (to appear). Google Scholar

[3] 3.Auinger, K., Free objects in joins of strict inverse and completely simple semigroup varieties, J. London Math. Soc. (to appear). Google Scholar

[4] 4.Grätzer, G., Universal Algebra (Van Nostrand, Princeton, 1978). Google Scholar

[5] 5.Grätzer, G., General Lattice Theory (Birkhäuser, Basel, 1978). Google Scholar | DOI

[6] 6.Hall, T. E., Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc. 40 (1989), 59–77. Google Scholar | DOI

[7] 7.Hall, T. E., Regular semigroups: amalgamation and the lattice of existence varieties, Algebra Universal 28 (1991), 79–108. Google Scholar | DOI

[8] 8.Hall, T. E., A concept of variety for regular semigroups; in Semigroup Theory, Proceedings of the Monash University Conference on Semigroup Theory in Honor of G. B. Preston, eds. Hall, T. E., Meakin, J. C. and Jones, P. R. (World Scientific, Singapore, 1991), 101–116. Google Scholar | DOI

[9] 9.Hall, T. E., On regular semigroups whose idempotents form a subsemigroup, Bull. Austral. Math. Soc. 1 (1969), 195–208. Google Scholar | DOI

[10] 10.Harary, F., Graph Theory (Addison-Wesley, 1971). Google Scholar

[11] 11.Howie, J. M., An Introduction to Semigroup Theory. (Academic Press, London, 1976). Google Scholar

[12] 12.Kaďourek, J. and Szendrei, M. B., A new approach in the theory of orthodox semigroups, Semigroup Forum 40 (1990), 257–296. Google Scholar | DOI

[13] 13.Lallement, G., Demi-groupes réguliers, Ann. Math. Pura Appl. 77 (1967), 47–129. Google Scholar | DOI

[14] 14.Lallement, G. and Petrich, M., Structure d'une classe de demi-groupes réguliers, J. Math. Pures Appl. 48 (1969), 345–397. Google Scholar

[15] 15.Margolis, S. W., Meakin, J. C. and Stephen, J. B., Free objects in certain varieties of inverse semigroups, Canad. J. Math. 42 (1990), 1084–1097. Google Scholar | DOI

[16] 16.Nordahl, T. E., and Scheiblich, H. E., Regular *-semigroups, Semigroup Forum 16 (1978), 369–377. Google Scholar | DOI

[17] 17.Petrich, M., Regular semigroups satisfying certain conditions on idempotents and ideals, Trans. Amer. Math. Soc. 170 (1972), 245–267. Google Scholar | DOI

[18] 18.Petrich, M., Inverse Semigroups (Wiley, New York, 1984). Google Scholar

[19] 19.Reilly, N. R., Free combinatorial strict inverse semigroups, J. London Math. Soc. (2) 39 (1989), 102–120. Google Scholar | DOI

[20] 20.Szendrei, M. B., Free *-orthodox semigroups, Simon Stevin 59 (1985), 175–201. Google Scholar

[21] 21.Szendrei, M. B., A new interpretation of free orthodox and generalized inverse *-semigroups; in Semigroups, Theory and Applications, Proceedings of the 1986 Oberwolfach Conference on Semigroups, eds. Lallement, G., Jürgensen, H. and Weinert, H., Lecture Notes in Mathematics 1320 (Springer-Verlag, 1988), 358–371. Google Scholar | DOI

[22] 22.Yeh, Y. T., The existence of e-free objects in e-varieties of regular semigroups, Int. J. Alg. Comput. (to appear). Google Scholar