Multi-Linear Iterative K-Σ-Semialgebras

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Abstract

We consider K-semialgebras for a commutative semiring K that are at the same time Σ-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear K-Σ-semialgebra. Examples of such algebras include the algebras of Σ-tree series over an alphabet A with coefficients in K, and the algebra of all rational tree series. We show that for many commutative semirings K, the rational Σ-tree series over A with coefficients in K form the free multi-linear iterative K-Σ-semialgebra on A.

Keywords

Semialgebra
Σ-algebra
rational tree-series
free algebra
unique fixed points

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1

Partially supported by the project TÁMOP-4.2.1/B-09/1/KONV-2010-0005 “Creating the Center of Excellence at the University of Szeged”, supported by the European Union and co-financed by the European Regional Fund, and by grant no. K 75249 from the National Foundation of Hungary for Scientific Research.