A note on two congruences on a groupoid

Let S be a groupoid and θ p , θ m {\theta _p},{\theta _m} the congruences on S defined as follows: x θ p y ( x θ m y ) 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 θ p {\theta _p} is the smallest congruence on S for which the quotient is a semilattice. It is also shown that S / θ m 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 θ p {\theta _p} and θ m {\theta _m} are defined on a lattice using lattice-ideals; quotients under these are distributive lattices.