A note on two congruences on a groupoid

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.