×

Commutativity of operators on the lattice of existence varieties. (English) Zbl 0870.20042

A class of regular semigroups is called an e-variety if it is closed under direct products, homomorphic images and regular subsemigroups. The article continues earlier work of the authors [J. Algebra 178, No. 3, 733-759 (1995; Zbl 0842.20051); Semigroup Forum 53, No. 1, 1-24 (1996; Zbl 0854.20069)]. In the former article, the authors introduced 7 complete congruences on the lattice \({\mathbf L}\) of e-varieties of regular semigroups. In the work under review, 4 new complete congruences on \({\mathbf L}\) are introduced. They have the form \(\alpha_{\mathcal P}\): \(\mathcal A\alpha_{\mathcal P}\mathcal B\Leftrightarrow\mathcal A\cap\mathcal P=\mathcal B\cap\mathcal P\) where \(\mathcal P\) is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. To each complete congruence \(\rho\) on \({\mathbf L}\) is associated an idempotent operator \({\mathcal A}\rightarrow{\mathcal A}^{\rho}\) on \({\mathbf L}\). Numerous results concerning the commutativity of such operators are established.

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
20M17 Regular semigroups
08A30 Subalgebras, congruence relations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Auinger, K.: The word problem for the bifree combinatorial strict regular semigroup. Math. Proc. Camb. Phil. Soc.113, 519-533 (1993). · Zbl 0798.20053 · doi:10.1017/S0305004100076179
[2] Auinger, K.: Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox*-semigroup varieties. Glasgow Math. J.35, 25-37 (1993). · Zbl 0779.20036 · doi:10.1017/S0017089500009538
[3] Auinger, K.: Free strict inverse semigroups. J. Algebra157, 26-42 (1993). · Zbl 0782.20047 · doi:10.1006/jabr.1993.1088
[4] Auinger, K.: The bifree locally inverse semigroup on a set. J. Algebra166, 630-650 (1994). · Zbl 0806.20052 · doi:10.1006/jabr.1994.1169
[5] Clifford, A. H., Preston, G. B.: The Algebraic Theory of Semigroups. Math. Surveys No. 7. Providence, RI: Amer. Math. Soc. Vol. I, 1961; Vol. II, 1967. · Zbl 0111.03403
[6] Doyle, J.: On existence varieties of orthodox semigroups. J. Austral. Math. Soc.58, 100-125 (1995). · Zbl 0828.20043 · doi:10.1017/S1446788700038131
[7] Gr?tzer, G.: Universal Algebra, 2nd ed. New York: Springer. 1979.
[8] Hall, T. E.: On regular semigroups whose idempotents from a subsemigroup. Bull. Austral. Math. Soc.1, 195-208 (1969). · Zbl 0172.31101 · doi:10.1017/S0004972700041447
[9] Hall, T. E.: On regular semigroups. J. Algebra24, 1-24 (1973). · Zbl 0262.20074 · doi:10.1016/0021-8693(73)90150-6
[10] Hall, T. E.: Some properties of local subsemigroups inherited by larger subsemigroups. Semigroup Forum25, 35-49 (1982). · Zbl 0497.20049 · doi:10.1007/BF02573586
[11] Hall, T. E.: Identities for existence varieties of regular semigroups. Bull. Austral. Math. Soc.40, 59-77 (1989). · Zbl 0666.20028 · doi:10.1017/S000497270000349X
[12] Hall, T. E.: Regular semigroups: amalgamation and the lattice of existence varieties. Algebra Universalis28, 79-102 (1991). · Zbl 0727.20041 · doi:10.1007/BF01190413
[13] Hall, T. E., Jones, P. R.: On the lattice of varieties of bands of groups. Pacific. J. Math.91, 327-337 (1980). · Zbl 0419.20043
[14] Howie, J. M.: An Introduction to Semigroup Theory. London: Academic Press. 1976. · Zbl 0355.20056
[15] Jones, P. R.: On the lattice of varieties of completely regular semigroups. J. Austral. Math. Soc.35, 227-235 (1983). · Zbl 0537.20030 · doi:10.1017/S1446788700025726
[16] Jones, P. R.: Mal’cev products of varieties of comppletely regular semigroups. J. Austral. Math. Soc.42, 227-246 (1987). · Zbl 0613.20038 · doi:10.1017/S1446788700028226
[17] Jones, P. R., Trotter, P. G.: Semidirect products of regular semigroups. Trans. Amer. Math. Soc. (to appear). · Zbl 0892.20037
[18] Kadourek, J., Szendrei, M. B.: A new approach in the theory of orthodox semigroups. Semigroup Forum40, 257-296 (1990). · Zbl 0705.20052 · doi:10.1007/BF02573274
[19] Margolis, S. W., Meakin, J. C.:E-unitary inverse monoids and the Cayley graph of a group presentation. J. Pure Appl. Algebra58, 45-76 (1989). · Zbl 0676.20037 · doi:10.1016/0022-4049(89)90052-2
[20] Meakin, J. C., Namboodripad, K. S. S.: Coextensions of pseudo-inverse semigroups by rectangular bands. J. Austral. Math. Soc.30, 73-86 (1980). · Zbl 0449.20063 · doi:10.1017/S1446788700021935
[21] Pastijn, F., Petrich, M.: Congruences on regular semigroups. Trans. Amer. Math. Soc.295, 607-633 (1986). · Zbl 0599.20095 · doi:10.1090/S0002-9947-1986-0833699-3
[22] Pastijn, F., Petrich, M.: The congruence lattice of a regular semigroup. J. Pure Appl. Algebra53, 93-123 (1988). · Zbl 0649.20052 · doi:10.1016/0022-4049(88)90015-1
[23] Pastijn, F., Petrich, M.: Congruence lattices on a regular semigroup associated with certain operators. Acta Sci. Math.55, 229-247 (1991). · Zbl 0797.20056
[24] Pastijn, F., Trotter, P. G.: Residual finiteness in completely regular semigroup varieties. Semigroup Forum37, 127-147 (1988). · Zbl 0627.20032 · doi:10.1007/BF02573130
[25] Petrich, M.: Inverse Semigroups. New York: Wiley. 1984. · Zbl 0546.20053
[26] Petrich, M., Reilly, N. R.: Near varieties of idempotent generated completely simple semigroups. Algebra Universalis16, 83-104 (1983). · Zbl 0516.20037 · doi:10.1007/BF01191755
[27] Petrich, M., Reilly, N. R.: Semigroups generated by certain operators on varieties of completely regular semigroups. Pacific J. Math.132, 151-175 (1988). · Zbl 0598.20061
[28] Petrich, M., Reilly, N. R.: Operators related to idempotent generated and monoid completely regular semigroups. J. Austral. Math. Soc.49, 1-23 (1990). · Zbl 0708.20019 · doi:10.1017/S1446788700030202
[29] Petrich, M., Reilly, N. R.: Completely Regular Semigroups. Book manuscript.
[30] Pol?k, L.: On varieties of completely regular semigroups. III. Semigroup Forum37, 1-30 (1988). · doi:10.1007/BF02573119
[31] Reilly, N. R.: Varieties of completely regular semigroups. J. Austral. Math. Soc.38, 372-393 (1985). · Zbl 0572.20040 · doi:10.1017/S144678870002365X
[32] Reilly, N. R.: Free combinatorial strict inverse semigroups. J. London Math. Soc.39, 102-120 (1989). · Zbl 0636.20032 · doi:10.1112/jlms/s2-39.1.102
[33] Reilly, N. R., Scheiblich, H. E.: Congruences on regular semigroups. Pacific J. Math.23, 349-360 (1967). · Zbl 0159.02503
[34] Reilly, N. R., Zhang, S.: Congruence relations on the lattice of existence varieties of regular semigroups. J. Algebra178, 733-759 (1995). · Zbl 0842.20051 · doi:10.1006/jabr.1995.1375
[35] Reilly, N. R., Zhang, S.: Operators and products in the lattice of existence varieties of regular semigroups. Semigroup Forum53, 1-24 (1996). · Zbl 0854.20069 · doi:10.1007/BF02574117
[36] Reilly, N. R., Zhang, S.: Associativity of products of existence varieties of regular semigroups. Preprint. · Zbl 0934.20044
[37] Trotter, P. G.: Congruence extensions in regular semigroups. J. Algebra137, 166-179 (1991). · Zbl 0714.20053 · doi:10.1016/0021-8693(91)90086-N
[38] Yamada, M.: On a certain class of regular semigroups. In: Proc. Symposium on regular semigroups. Northern Illinois University, 146-179 (1979). · Zbl 0446.20039
[39] Yeh, Y. T.: The existence of e-free objects in e-varieties of regular semigroups. Internat. J. Algebra Comput.2, 471-484 (1992). · Zbl 0765.20030 · doi:10.1142/S0218196792000281
[40] Yeh, Y. T.: On existence varieties ofE-solid or locally inverse semigroups and e-invariant congruences. J. Algebra164, 500-514 (1994). · Zbl 0811.20054 · doi:10.1006/jabr.1994.1072
[41] Zhang, S.: Completely regular semigroup varieties generated by Mal’cev products with groups. Semigroup Forum48, 180-192 (1994). · Zbl 0797.20048 · doi:10.1007/BF02573668
[42] Zhang, S.: Certain operators related to Mal’cev products on varieties of completely regular semigroups. J. Algebra168, 249-272 (1994). · Zbl 0815.20050 · doi:10.1006/jabr.1994.1228
[43] Zhang, S.: An infinite order operator on the lattice of varieties of completely regular semigroups. Algebra Universalis35, 485-505 (1996) · Zbl 0860.20045 · doi:10.1007/BF01243591
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.