On the zero-divisor graph of the abian poset and complemented rings

The order-theoretic zero-divisor graphs of a reduced commutative ring R are considered with respect to certain extensions of the Abian order, and characterizations are given for when these graphs coincide with the usual ring-theoretic zero-divisor graph of R. For instance, it is proved that the Abian order induces a zero-divisor graph that is equal to the ring-theoretic zero-divisor graph if and only if R is a Boolean ring. Moreover, the ring and graph-theoretic definitions of complementation that have appeared in earlier work are contrasted with the usual order-theoretic notions of complementation and pseudocomplementation. For example, it is shown that R is complemented in the ring-theoretic sense (which is well known to be equivalent to R having a von Neumann regular total quotient ring) if and only if R can be pseudocomplemented in the order-theoretic sense by certain extensions of the Abian order.