Interpretation of Planar Ternary Ring in Desargues Plane Like Usual Ring

Abstract

Use of ``ring'' in the determine of a planar ternary ring seems unjustified at first sight. In this paper we show that in Desargues affine plane, in certain condition, planar ternary ring (S, t) turns into usually associative ring $(S, +, \cdot)$. So, in an affine plane $\mathpzc{A}=(\mathcal{P}, \mathcal{L}, \mathcal{I})$ coordinatized from a coordinative system (O, I, OX, OY, OI) and bijection $\sigma : (OI)\to S$, determine a ternary operation $t : S^{3} \to S$, which we call planar ternary operation. We prove that every coordinatizing affine plane $\mathpzc{A}=(\mathcal{P}, \mathcal{L}, \mathcal{I})$ determine a planar ternary ring (S, t), where S is the set of coordinates of affine plane and t its planar ternary operation, and vise versa. Also we introduce the binary operation of addition and multiplication in S and underline relations $a+b=t(a, 1, b)$, $a\cdot b=t(a, b, 0)$. Since affine plane is fulfilled with the first Desargues axiom D1, related to the structure $(S, +, \cdot)$ in that plane, considering some isomorphism imply that $(S, +)$ is abelian group. In the following, we show that when in an affine plane except first Desargues axiom D1 also hold the second Desargues axiom D2, then its planar ternary ring $(S, t)$ is the usual associative ring $(S, +, \cdot)$.