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A note on two congruences on a groupoid


Author: K. Nirmala Kumari Amma
Journal: Proc. Amer. Math. Soc. 65 (1977), 204-208
MSC: Primary 20L05
DOI: https://doi.org/10.1090/S0002-9939-1977-0444807-0
MathSciNet review: 0444807
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Abstract: Let S be a groupoid and $ {\theta _p},{\theta _m}$ the congruences on S defined as follows: $ x{\theta _p}y\;(x{\theta _m}y)$ iff every prime (minimal prime) ideal of S containing x contains y and vice versa. It is proved that $ {\theta _p}$ is the smallest congruence on S for which the quotient is a semilattice. It is also shown that $ S/{\theta _m}$ is a disjunction semilattice if S has 0 and is a Boolean algebra if S is intraregular and closed for pseudocomplements. Some connections between the ideals of S and those of the quotients are established. Congruences similar to $ {\theta _p}$ and $ {\theta _m}$ are defined on a lattice using lattice-ideals; quotients under these are distributive lattices.


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  • [1] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1948. MR 0029876 (10:673a)
  • [2] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. I, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R.I., 1961. MR 0132791 (24:A2627)
  • [3] J. E. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. 13 (1963), 31-50. MR 0143837 (26:1387)
  • [4] K. Nirmala Kumari Amma, Pseudocomplements in groupoids (to appear). MR 511605 (80c:20077)
  • [5] J. Varlet, Congruences dans les treillis pseudocomplémentes, Bull. Soc. Roy Sci. Liège 9 (1963), 623-635. MR 0159768 (28:2984)
  • [6] P. V. Venkatanarasimhan, A note on modular lattices, J. Indian Math. Soc. 30 (1966), 55-59. MR 0213267 (35:4131)
  • [7] -, Semi-ideals in semilattices, Colloq. Math. 30 (1974), 203-212. MR 0360388 (50:12838)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0444807-0
Keywords: Groupoid, intraregular, prime ideal, filter, congruence, semilattice disjunction property, pseudocomplement, Boolean algebra
Article copyright: © Copyright 1977 American Mathematical Society

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