The monoid of orientation-preserving mappings on a chain
Let \(X=\{1,2,\dots,n\}\) have the usual ordering. \(T_n\) is the full transformation semigroup on \(X\) and \(O_n\) is the submonoid of \(T_n\) consisting of all order preserving selfmaps of \(X\). A finite sequence \(\{a_1,a_2,\dots,a_k\}\) of elements of \(X\) is defined to be cyclic if there is at most one subscript \(i\) such that \(a_i>a_{i+1}\) and it is defined to be anti-cyclic if there is at most one subscript \(i\) such that \(a_i<a_{i+1}\). Note that a sequence can be both cyclic and anti-cyclic. An example is \((1,2,1)\). An element \(\alpha\in T_n\) is said to be orientation-preserving on \(X\) if \((1\alpha,2\alpha,\dots,n\alpha)\) is cyclic. The collection of all orientation-preserving maps on \(X\) is denoted by \(OP_n\) and is a submonoid of \(T_n\) which contains \(O_n\). The monoid \(OP_n\) is the main object of investigation in this paper and the authors obtain many varied results about this semigroup. For example they show that \(OP_n\) is regular and every subgroup of \(OP_n\) is cyclic. Let \(E(OP_n)\) denote the collection of idempotents of \(OP_n\) and let \(F_n\) denote the \(n\)th Fibonacci number. They show that \(| E(OP_n)|=F_{2n-1}+F_{2n+1}-(n^2-n+2)\) for all \(n\geq 1\). Let \(O^k_n=\{a^{-k}fa^k:f\in O_n\}\), \(0\leq k\leq n-1\), and let \(U=\bigcup^{n-1}_{k=0} O^k_n\). The subsemigroup of a semigroup \(S\) which is generated by a subset \(A\) is denoted by \(\langle A\rangle\). The authors show that \(\langle U\rangle=\langle E(OP_n)\rangle=(OP_n\setminus Z_n)\cup\{1_X\}\). A mapping \(\alpha\in T_n\) is defined to be orientation-reversing if the sequence \((1\alpha,2\alpha,\dots,n\alpha)\) is anti-cyclic and the collection of all orientation-reversing maps is denoted by \(OR_n\). It is shown that \(P_n=OP_n\cup OR_n\) is a regular submonoid of \(T_n\) and they go on to get additional results about this submonoid as well.
scientific article; zbMATH DE number 1251282