We analyse the classical property of centrality of equivalence relations in terms of normal monomorphisms. For this purpose, the internal structure of connector is introduced, allowing to clarify classical results in Maltsev categories and to prove new ones in protomodular categories. This approach allows to work in the general context of finitely complete categories, without requiring the usua...

An algebra is called subalternative if the associator of any three linearly dependent elements is their linear combination. We prove that in characteristic 6 = 2; 3 any such algebra is Maltsev{admissible and can be identiied with a hyperplan in certain unital alternative algebra.

Meet semidistributive varieties are in a sense the last of most important classes universal algebra for which it is unknown whether can be characterized by strong Maltsev condition. We present new, relatively simple condition characterizing meet-semidistributive varieties, and provide candidate

We prove that for any variety V, the existence of an edge-term (defined in [1]) and Jónsson terms is equivalent to the existence of a near-unanimity term. We also characterize the idempotent Maltsev conditions which are defined by a system of linear absorption equations and which imply congruence distributivity.

Relationship is clariied between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear extensions by bifunctors from the so called abelian theories. The latter theories are described; they...

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We prove that the problem has a dichotomy in the class of finite groupoids with an identity element. By developing the underlying idea further, we present a dichotomy theorem in the class of finite algebras that admit a non-trivial idempotent Maltsev condition. Th...