Essential unifiers

A substitution δ is less general than a substitution σ if there exists λ   such that δ=σ⋅λδ=σ⋅λ, which induces a notion of generality in the algebra of substitutions. We propose to look at this well known concept of generality again, and to impose a new quasi ordering on substitutions as a natural result of a stronger notion of the composition of substitutions. This new generality ordering has important consequences for the theory of E-unification (unification in equational theories) and changes the basic notion of the most general unifiers, now called essential unifiers, as well as the unification hierarchy. In particular we show that for idempotent Semigroups (associativity and idempotency), also known as Bands, the set of essential unifiers always exists and is finite.