Directly decomposable congruences in varieties with nullary operations
It is well-known that every monounary variety of total algebras has one-element equational basis (see [5]). In my paper I prove that every monounary weak variety has at most 3-element equational basis. I give an example of monounary weak variety having 3-element equational basis, which has no 2-element equational basis.
We obtain the construction of free abelian extensions in a congurence-permutable variety V using the construction of a free abelian extension in a variety of algebras with one ternary Mal'cevoperation and a monoid of unary operations. We also use this construction to obtain a free solvable V-algebra.
An algebra is subregular alias regular with respect to a unary term function if for each we have whenever for each . We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset is a class of some congruence on containing if and only if is this generalized deductive system. This method is efficient (needs a finite number of steps).