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 | 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.

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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