Cyclic subsemigroups of symmetric inverse semigroups
Let \(X=\{1,2,...,n\}\). A partial one-one transformation on X is referred to as a chart and the symmetric inverse semigroup on X (i.e., the semigroup of all charts on X) is denoted by \(C_ n\). For \(a\in C_ n\) and \(\{i_ j\}^ k_{j=1}\subseteq X\), suppose \(i_ ja=i_{j+1}\) for \(1\leq j\leq k-1\). Then a is called a path. If \(i_ ka\neq i_ 1\), then a is called a proper path of length \(k+1\) and if \(i_ k=i_ 1\), a is referred to as a circuit of length k. Let \(D_ a\) and \(R_ a\) denote, respectively, the domain and range of a. If \((D_ a\cup R_ a)\cap (D_ b\cup R_ b)=\emptyset\), the chart c defined by \(xc=xa\) for \(x\in D_ a\) and \(xc=xb\) for \(x\in D_ b\) is referred to as the join of a and b and a and b are said to be disjoint. The join of a and b, when it exists, is denoted by ab \((=ba)\). It is appropriate to emphasize the fact that the symbol ab does not, in this paper, represent the product of the transformations a and b. Of course, the product of a and b always exists. The join may not but if it does, the product will be the empty transformation. The author proves two fundamental results about charts. The first states that every chart in \(C_ n\) is the join of disjoint paths and the second states that if a is any chart and \(a=b_ 1b_ 2...b_ t\) is a join representation of paths each of length at least 2, then the representation is unique except for the order of the paths. He goes on to show that for \(a\in C_ n-S_ n\) (where \(S_ n\) is the permutation group on X), the index of \(\), the cyclic subsemigroup of \(C_ n\) generated by a, is the maximum of the lengths of the proper paths in the join representation of a and the period of a is the least common multiple of the lengths of the circuits that appear in the join representation of a (if no circuits appear, the period is one). Thus, one can tell by inspecting the join representations of two charts whether or not the cyclic subsemigroups they generate are isomorphic.
scientific article; zbMATH DE number 3977285