Generalized semigroups of quotients
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- by C. V. Hinkle PDF
- Trans. Amer. Math. Soc. 183 (1973), 87-117 Request permission
Abstract:
For S a semigroup with 0 and ${M_S}$ a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.References
- Keizo Asano, Γber die Quotientenbildung von Schiefringen, J. Math. Soc. Japan 1 (1949), 73β78 (German). MR 31468, DOI 10.2969/jmsj/00120073 Gordon L. Bailes, Right inverse semigroups, Thesis, Clemson University, Clemson, S.C., 1972.
- P. Berthiaume, The injective envelope of $S$-sets, Canad. Math. Bull. 10 (1967), 261β273. MR 213457, DOI 10.4153/CMB-1967-026-1
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR 0218472
- E. H. Feller and R. L. Gantos, Indecomposable and injective $S$-systems with zero, Math. Nachr. 41 (1969), 37β48. MR 254164, DOI 10.1002/mana.19690410104
- T. E. Hall, On regular semigroups whose idempotents form a subsemigroup, Bull. Austral. Math. Soc. 1 (1969), 195β208. MR 249527, DOI 10.1017/S0004972700041447 C. V. Hinkle, Jr., The extended centralizer of an S-set (submitted).
- R. E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891β895. MR 45695, DOI 10.1090/S0002-9939-1951-0045695-9 F. R. McMorris, On quotient semigroups (submitted).
- F. R. McMorris, Vital injective $S$-systems, Math. Nachr. 47 (1970), 121β125. MR 283109, DOI 10.1002/mana.19700470114
- F. R. McMorris, The singular congruence and the maximal quotient semigroup, Canad. Math. Bull. 15 (1972), 301β303. MR 310101, DOI 10.4153/CMB-1972-056-8 Mario Petrich, Topics in semigroups, Pennsylvania State University, 1967.
- David A. Smith, On semigroups, semirings, and rings of quotients, J. Sci. Hiroshima Univ. Ser. A-I Math. 30 (1966), 123β130. MR 207879
- Yuzo Utumi, On quotient rings, Osaka Math. J. 8 (1956), 1β18. MR 78966
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 183 (1973), 87-117
- MSC: Primary 20M10
- DOI: https://doi.org/10.1090/S0002-9947-1973-0335666-6
- MathSciNet review: 0335666