Partial Algebras and Implications of (Weak) Matrix Properties

Matrix properties are a type of property of categories which includes the ones of being Mal’tsev, arithmetical, majority, unital, strongly unital, and subtractive. Recently, an algorithm has been developed to determine implications \(\textrm{M}\Rightarrow _{\textrm{lex}_*}\textrm{N}\) between them. We show here that this algorithm reduces to constructing a partial term corresponding to \(\textrm{N}\) from a partial term corresponding to \(\textrm{M}\). Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.