Commutativity preservers of incidence algebras

Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We describe the bijective linear maps $\varphi:I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $\varphi(D(X,K))=D(X,K)$. We prove that such a map $\varphi$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(\theta,\sigma,c,\kappa)$ of simpler maps $\theta$, $\sigma$, $c$ and a sequence $\kappa$ of elements of $K$

Local/Non-Local Complementarity in Topological Effects

In certain topological effects the accumulation of a quantum phase shift is accompanied by a local observable effect. We show that such effects manifest a complementarity between non-local and local attributes of the topology, which is reminiscent but yet different from the usual wave-particle complementarity. This complementarity is not a consequence of non-commutativity, rather it is due to the non-canonical nature of the observables. We suggest that a local/non-local complementarity is a general feature of topological effects that are ``dual'' to the AB effect.Comment: 4 page