On Projection Algebras
by
M. Mahmoudi
Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran.

The notion of a projection space (algebra) was first introduced in 1987 by Ehrig (&, ... ) as an algebraic version of ultrametric spaces. Computer Scientists use this notion for a formal description of parallel concurrent systems. One of the main problem in this scope is the specification of infinite objects (processes) which can not be denoted by finite terms. So, they use projection algebras as a convenient mean for algebraic specification of process algebras.

A projection space (algebra) is in fact a set with an action of a monoid M=\N^\infty = \N \cup {\infty}, where \N is the set of natural numbers and n < \infty,   for all n in \N with the binary operation m.n=min{ m, n}, on it.

In other words, a projection algebra is a set A together with a family (\lambda_n)_{n in \N^\infty} of unary operations \lambda_n:A --> A (called projections) such that

\lambdam o \lambdan=\lambdan.m and \lambda\infty = idA

for every m, n in \N^\infty. We denote \lambda_n(a) by an, for n in \N^\infty ,  a in A.

Some algebraic notions such as purity, equational compactness, tensor products, flatness, and weekly flatness have been studied for projection algebras. Here, injectivity and Baer Criterion for projection algebras is considered.

Date received: January 31, 2003