Cardinal and ordinal arithmetics of $n$-ary relational systems and $n$-ary ordered sets

Abstract: The aim of this paper is to define and study cardinal (direct) and ordinal operations of addition, multiplication, and exponentiation for $n$-ary relational systems. $n$-ary ordered sets are defined as special $n$-ary relational systems by means of properties that seem to suitably generalize reflexivity, antisymmetry, and transitivity from the case of $n=2$ or 3. The class of $n$-ary ordered sets is then closed under the cardinal and ordinal operations.

Keywords: $n$-ary relational system, $n$-ary ordered set, cardinal sum, cardinal product, cardinal power, ordinal sum, ordinal product, ordinal power

Classification (MSC2000): 04A05, 06A99, 08A02

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