Some Nuances of Many-sorted Universal Algebra: A Review (original) (raw)

2011, Bulletin of the EATCS

proof of the results. We give some examples of this, indicating how equational calculus, Birkhoff's variety theorem and interpolation results should be adjusted for many-sorted algebras. 2 Basic Definitions and Facts 2.1 Many-sorted Sets Let S be any set; we think of elements of S as sort names, or sorts for short. An S-sorted set is an S-indexed family of sets X = X s s∈S. We say that such an S-sorted set X is empty if X s is empty for all s ∈ S. The empty S-sorted set will be written (ambiguously) as ∅. We say that X is everywhere non-empty if X s ∅ for all s ∈ S ; otherwise we say that X is somewhere empty. Clearly, if S has at least two elements, there are S-sorted sets that are neither empty nor everywhere non-empty. S-sorted set X is finite if X s is finite for all s ∈ S and X s = ∅ for almost all s ∈ S (that is, for all but a finite number of s ∈ S , X s = ∅). Let X = X s s∈S and Y = Y s s∈S be S-sorted sets. Union, intersection, Cartesian product, disjoint union, inclusion (subset) and equality of X and Y are defined component-wise in the obvious manner. An S-sorted function f : X → Y is an S-indexed family of functions f = f s : X s → Y s s∈S ; X is called the domain (or source) of f , and Y is called its codomain (or target). An S-sorted function f : X → Y is an identity (inclusion, surjection, injection, bijection, etc) if for every s ∈ S , the function f s : X s → Y s is an identity (inclusion, surjection, injection, bijection, etc). The identity S-sorted function on X will be written as id X : X → X.