DEFINITION OF PARTIALITY
Leszek Rudak · Demonstratio Mathematica · 1987
We call definition of partiality a condition whioh is satisfied in a class of partial algebras if the class contains no nontrivial total algebras.In other words one can say that the condition of partiality is a condition, which cannot be satisfied by a nontrivial total algebra.The aim of this paper is to formulate the definition of partiality for a variety (i.e. an equationally definable class) of partial algebras.We introduce this definition in the form of a Mal'cev oondition.When dealing with equations in partial algebras one must decide what kind of equations will be considered.We have ohosen so called weak equations.For strong and existence equations our problem in easily solvable (see end of this paper).A signature is a pair with F a set of operation symbols and n a mapping from F into the set of nonnegative integers.A * (A,(f~:fe ?)) is a partial algebra of signature if A is a nonempty set and f-a partial n(f)-ary operation in A. Fix a signature .In the following we deal with partial algebras of signature only and thus the term "algebra" will always be used in the sense "partial algebra of signature