On Surjective Kernels of Partial Algebras

A partial algebra A = (A, F ) is called surjective if each of its elements lies in the range of some of its operations. By a transfinite iteration construction over the class of all ordinals it is proved that in each partial algebra A there exists the largest surjective subalgebra Skr A, called the surjective kernel of A. However, what might be found a bit surprising, for each ordinal α there is an algebra A with only finitary operations (even with a single unary operation), such that the described construction stops exactly in α steps. The result is compared with the classical ones on perfect kernels of first countable topological spaces. We use standard set-theoretical notation and terminology; in particular Y X denotes the set of all functions from the set X into the set Y , each ordinal α is represented as the set of all ordinals β < α, the least ordinal of cardinality אγ is denoted by ωγ , and ω = ω0. Under the term “partial algebra” we will understand a pair A = (A,F ), where A is an arbitrary set and F is a set of partial (finitary or infinitary) operations on A (we do not exclude any of the possibilities A = ∅ or F = ∅). For an operation f ∈ F we denote by ar(f) the arity and by D(f) the domain of f . This is to say that to each f ∈ F two sets ar(f) (in most cases ar(f) is assumed to be an ordinal) and D(f) ⊆ A are assigned, such that f : D(f) → A. A partial algebra A = (A,F ) will be called finitary if ar(f) is finite for each f ∈ F . A will be called a total algebra, or simply an algebra if all its operations are total, i.e., D(f) = A for each f ∈ F . Any subset B ⊆ A closed with respect to all operations f ∈ F , i.e. f(b) ∈ B whenever b ∈ D(f)∩B, will be called a subalgebra of A and it will be identified with the corresponding partial algebra B = (B,FB), where FB = {fB; f ∈ F} and fB denotes the restriction of f to B, i.e. ar(fB) = ar(f), D(fB) = D(f) ∩ B and fB(b) = f(b) for b ∈ D(fB). Obviously, every subalgebra of a total algebra is total, as well. Received November 1, 1991; revised February 19, 1992. 1980 Mathematics Subject Classification (1991 Revision). Primary 08A05, 08A55, 08A62, 08A65; Secondary 08A30, 08A60, 05C05, 54A20.