Automorphisms of lexicographic products

The automorphism group Γ(P) of a partially ordered set P consists of all permutations on P that preserve order (and have order preserving inverses). In this paper we raise, and partially answer, the question: How is the automorphism group of the lexicographic product (P × Q) of two orders (P and Q) related to the automorphism groups of the factors, P and Q?We show that the wreath product Γ(Q) wr ΓP) is always contained in Γ(P × Q). We find necessary and sufficient conditions for Γ(P × Q) to equal Γ(Q) wr Γ(P), and when P and Q are finite, we find a complete characterization of Γ(P × Q) in terms of Γ(P), Γ(Q) and properties of P and Q.