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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *S* be a groupoid and the congruences on *S* defined as follows: iff every prime (minimal prime) ideal of *S* containing *x* contains *y* and vice versa. It is proved that is the smallest congruence on *S* for which the quotient is a semilattice. It is also shown that 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 and are defined on a lattice using lattice-ideals; quotients under these are distributive lattices.

**[1]**Garrett Birkhoff,*Lattice Theory*, American Mathematical Society Colloquium Publications, vol. 25, revised edition, American Mathematical Society, New York, N. Y., 1948. MR**0029876****[2]**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****[3]**Joseph Kist,*Minimal prime ideals in commutative semigroups*, Proc. London Math. Soc. (3)**13**(1963), 31–50. MR**0143837**, https://doi.org/10.1112/plms/s3-13.1.31**[4]**K. Nirmala Kumari Amma,*Pseudocomplements in groupoids*, J. Austral. Math. Soc. Ser. A**26**(1978), no. 2, 209–219. MR**511605****[5]**Jules Varlet,*Congruences dans les treillis pseudo-complémentes*, Bull. Soc. Roy. Sci. Liège**32**(1963), 623–635 (French). MR**0159768****[6]**P. V. Venkatanarasimhan,*A note on modular lattices*, J. Indian Math. Soc. (N.S.)**30**(1966), 55–59 (1967). MR**0213267****[7]**P. V. Venkatanarasimhan,*Semi-ideals in semi-lattices*, Colloq. Math.**30**(1974), 203–212. MR**0360388**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20L05

Retrieve articles in all journals with MSC: 20L05

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