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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
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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Keywords: Groupoid, intraregular, prime ideal, filter, congruence, semilattice disjunction property, pseudocomplement, Boolean algebra
Article copyright: © Copyright 1977 American Mathematical Society

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