Abstract
The aim of this paper is to define the notion of pseudo-valuations on quantum B-algebras and to investigate their properties. We also give characterizations of pseudo-valuations and we show that there is a relationship between the positive strong pseudo-valuations and the filters of a unital quantum B-algebra. We define and characterize the pseudo-valuations on the particular cases of quantum B-algebras with product and pointed quantum B-algebras. We also prove extension theorems for pseudo-valuations on good quantum B-algebras satisfying Glivenko property, and we investigate certain possible generalizations of Horn-Tarski extension theorem for the case of pseudo-valuations on quantales.
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