On representation type of a pair of posets with involution

Abstract

In this paper we consider the problem on classifying the representations of a pair  of posets with involution. We prove that if  one of these is a chain of length at least 4 with trivial involution and the other is with nontrivial one, then the pair is tame  \(\Leftrightarrow\) it is of finite type \(\Leftrightarrow\) the poset with nontrivial involution is a \(*\)-semichain (\(*\) being the involution). The case that each of the posets with involution is not a chain with trivial one was considered by the author earlier. In proving our result we do not use the known technically difficult results  on representation type of posets with involution.