Cartesian Closedness in Cat - Egories of Partial

Keywords: Exponential subcategory of a category, cartesian closed category, initially structured category, category of partial algebras of the same type, partial algebra fulllling the interchange law, diagonal partial algebra. Abstract: We study categories of partial algebras of the same type. In these categories we deene a binary operation of exponentiation for objects and investigate its behaviour. We discover two cartesian closed initially structured subcategories in every category of partial algebras of the same type. It is well known that concrete categories having well-behaved function spaces, i.e. being initially structured and cartesian closed, play an important role in applications to many branches of mathematics. It is therefore worthwhile to look for such categories also among categories of general algebraic systems. In this note we focus our interest onto categories of partial algebras. As for generality, partial algebras lie between total (i.e. universal) algebras and relational systems. Therefore, when studying partial algebras, we can extend considerations known for total algebras or restrict those known for relational systems. However, such an extension or restriction is often not quite trivial and many new particular considerations have to be done for partial algebras.