Narrow Results

This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

This paper provides an abstract characterization of the monoids that appear as monoids of subalgebras of the square of finite minimal algebras admitting a majority term. As a tool, a matrix construction similar to the one introduced in [A characterization of the inverse monoid of bi-congruences of certain algebras, Int. J. Algebra Comput.6 (2009) 791–808] is used.

This paper studies nilpotent and Hamiltonian cancellative residuated lattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular, results about lattice-ordered groups are extended to the domain of residuated lattices. The two key ingredients that underlie the considerations of this paper are the categorical equivalence between Ore residuated lattices and lattice-ordered groups endowed with a suitable modal operator; and Malcev’s description of nilpotent groups of a given nilpotency class c in terms of a semigroup equation.

We reduce the problem of categorical equivalence for finite rings to the case of rings of prime power characteristics. It is proved that categorically equivalent rings of coprime characteristics must be semisimple. The categorical equivalence problem for finite semisimple rings is completely solved.

In this paper, we showed that any clone of operations preserving a nontrivial $n$-equivalence on a finite set is categorically equivalent to the clone of operations preserving a nontrivial $n$-equivalence on a set having the cardinality of $3$.