Commutative semigroups satisfying the equation $2x+y=2x$ and having only two $G$-invariant congruences for an automorphism group $G$ are considered. Some classes of these semigroups are characterized and some other examples are constructed.
The study of paramedial groupoids (with emphasis on the structure of simple paramedial groupoids) was initiated in [1] and continued in [2], [3] and [5]. The aim of the present paper is to give a full description of finite simple zeropotent paramedial groupoids (i.e., of finite simple paramedial groupoids of type (II)—see [2]). A reader is referred to [1], [2], [3] and [7] for notation and various prerequisites.
The paper is an immediate continuation of [3], where one can find various notation and other useful details. In the present part, a full classification of infinite simple zeropotent paramedial groupoids is given.
For every module $M$ we have a natural monomorphism \[ \Psi :\coprod _{i\in I}\mathop {\mathrm Hom}\nolimits _R(M,A_i)\rightarrow \mathop {\mathrm Hom}\nolimits _R\biggl (M,\coprod _{i\in I}A_i\biggr) \] and we focus our attention on the case when $\Psi $ is also an epimorphism. Some other colimits are also considered.
By a paramedial groupoid we mean a groupoid satisfying the equation ax·yb=bx·ya. This equation is, in certain sense, symmetric to the equation of mediality xa·by=xb·ay and, in fact, the theories of both varieties of groupoids are parallel. The present paper, initiating the study of paramedial groupoids, is meant as a modest contribution to the enormously difficult task of describing algebraic properties of varieties determined by strong linear identities (and, especially,of the corresponding simple algebras).
This short note is a continuation of and and its purpose is to show that every simple zeropotent paramedial groupoid containing at least three elements is strongly balanced in the sense of [4].
We prove that the semirings of 1-preserving and of 0,1-preserving endomorphisms of a semilattice are always subdirectly irreducible and we investigate under which conditions they are simple. Subsemirings are also investigated in a similar way.