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How Iterative are Iterative Algebras? J. Adamek 1 , S. Milius and J. Velebil 1

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,
extensive category
1991 MSC: 68Q65, 18A15
1 Introduction
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


Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig


Collections: Computer Technologies and Information Sciences