Certain congruences on free trioids

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. This paper is devoted to the study of congruences on trioids. We characterize the least dimonoid congruences and the least semigroup congruence on the free (commutative, rectangular) trioid.     

Rectangular Ring Congruences on an E-inversive Semiring

In this paper, we discussed the property of rectangular band semiring congruence and ring congruence on a semiring and gave some characterizations and structure of rectangular ring congruence on an E-inversive semiring.     

E-inversive semigroups with a completely simple kernel

We show that an E-inversive semigroup S has a completely simple kernel K_S if and only if it contains a primitive idempotent (in that case, K_S is the set-theoretic union of the groups eSe, where e is a primitive idempotent of S). Along the way, some equivalent conditions for a semigroup to be E-inversive are given. Moreover, some applications of the above theorem will be pointed out.     

Compact factor congruences imply Boolean factor congruences

We prove that any variety in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.     

Some congruences on trioids

We present the least idempotent congruence on the trioid with a commutative operation, the least semilattice congruence on the trioid with an idempotent operation, and the least separative congruence on the trioid with a commutative operation. Also we construct different examples of trioids.     

A note on quadratic congruences

The notion of quadratic congruences was introduced recently in [1]. In the present note we revise the original definition and present an explanation of the the phenomena.     

Monomial congruences

Letn ₁,n ₂,...,n _t integers. It is proved that the monomial congruence is solvable for allm2 and (a, m)=1 if and only if (n ₁ ,n ₂ ..., n _t )=1.     

Complemented congruences on Ockham algebras

An Ockham algebra that satisfies the identity is called a K_{n, m}-algebra. Generalizing some results obtained in [2], J. Varlet and T. Blyth, in [3, Chapter 8], study congruences on K_{1, 1}-algebras. In particular, they describe the complement (when it exists) of a principal congruence and characterize these congruences that are complemented. In this paper we study the same question for K_{n, m}-algebras. Received March 24, 2005; accepted in final form April 28, 2005.