Summary: How Iterative are Iterative Algebras?
J. Ad’amek 1 , S. Milius and J. Velebil 1
Iterative algebras are defined by the property that every guarded system of recursive
equations has a unique solution. We prove that they have a much stronger property:
every system of recursive equations has a unique strict solution. And we characterize
those systems that have a unique solution in every iterative algebra.
Key words: iterative algebra, guarded equation, strict solution,
1991 MSC: 68Q65, 18A15
The aim of the present paper is to show that iterative algebras, i.e. algebras
with unique solutions of all guarded systems of recursive equations, have so
lutions of unguarded systems as well. In fact, we introduce a natural concept
of a``strict'' solution (which is one that assigns to every ungrounded variable
the result #) and prove that iterative algebras have unique strict solutions of
all systems of recursive equations.
The motivation for our paper is twofold. Firstly, in the paper of Evelyn
Nelson [N] which introduced iterative algebras as a means to study the it
erative theories of Calvin Elgot [E] (see also a very similar concept of Jerzy