NEW STRUCTURES IN PSEUDO MAGIC SQUARES

Abstract

A pseudo magic square (PMS) of order n is an n×n square matrix whose entries are integers such that the sum of the numbers of any row and any column is the same number, the magic constant. It is a generalization of the concept of magic squares. In this paper we investigate new algebraic structures of PMS’s. We explore the group structure of PMS’s to show that the quotient of the group of PMS’s of order n by its subgroup with zero constant is isomorphic to the infinite additive group of integers, where theisomorphism is constructed by means of the magic constants of the corresponding PMS’s. We investigate the ring structure of PMS’s to characterize nilpotent and idempotent PMS’s as well as we show that the set of PMS’s of zero constant is a two-sided ideal in the ring of PMS’s. Thus, we can define the quotient ring of PMS’s. Moreover, we introduce an invariant and a weak invariant of PMS’s and show some results derived from such definitions. In particular, we show that the set of weak invariants of PMS’s forms a Z-module under the pointwise addition and scalar multiplication. AMS Subject Classification: 15B36