On the gravity function of singular transformations: characterizations and structural properties

Abstract

Let \(\mathcal{T}_n\) denote the full transformation semigroup on an \(n\)-element set, and let \(\operatorname{Sing}_n\) be its subsemigroup of singular mappings. For \(\alpha \in \operatorname{Sing}_n\), the gravity \(g(\alpha) = n + c(\alpha) - f(\alpha)\) measures the minimum number of defect-one idempotents required to express \(\alpha\) as a product, where \(c(\alpha)\) is the number of cyclic orbits of length at least two and \(f(\alpha)\) is the number of fixed points. In this paper, we completely characterize when a singular mapping satisfies \(g(\alpha) = n\) and when \(g(\alpha) = s(\alpha)\), where \(s(\alpha) = n - f(\alpha)\) is the rank defect. We prove that \(g(\alpha) = n\) if and only if \(c(\alpha) = f(\alpha)\), and \(g(\alpha) = s(\alpha)\) if and only if \(\alpha\) has no cyclic orbits. Consequently, the identity \(g(\alpha) - s(\alpha) = c(\alpha)\) provides a direct combinatorial interpretation of the difference between these invariants. We provide a complete enumeration of idempotents satisfying \(g(\alpha) = s(\alpha)\), obtaining the classical count \(\sum_{k=1}^{n} \binom{n}{k} k^{n-k}\), with the singular case obtained by excluding the identity mapping. We analyze the distribution of these properties within group \(\mathcal{H}\)-classes of the semigroup, demonstrating that for any group \(\mathcal{H}\)-class in a \(\mathcal{D}\)-class \(D_k\), the sets of elements satisfying \(g(\alpha) = n\) and \(g(\alpha) = s(\alpha)\) are disjoint subsets of size at most \(k!\). Additionally, we establish that the classes of cycle-free mappings are closed under composition, with the property that \(g(\alpha\beta) = s(\alpha\beta)\) for all elements in these classes. These results offer new insights into the relationship between the orbit structure and the idempotent-generated nature of singular transformations.