On monoids of injective partial selfmaps almost everywhere the identity

In this paper we study the semigroup $\mathscr{I}^{\infty}\lambda$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $\lambda$. We describe the Green relations on $\mathscr{I}^{\infty}\lambda$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}\lambda$. We prove that every Hausdorff hereditary Baire topology $\tau$ on $\mathscr{I}^{\infty}\omega$ such that $(\mathscr{I}^{\infty}\omega,\tau)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}\lambda$ in a topological semigroup. Also we show that for an infinite cardinal $\lambda$ the discrete semigroup $\mathscr{I}^{\infty}\lambda$ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning $\mathscr{I}^{\infty}\lambda$ into a topological inverse semigroup.

💡 Research Summary

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The paper investigates the semigroup (\mathscr{I}^{\infty}{\lambda}) consisting of all injective partial self‑maps of a set (X) of infinite cardinality (\lambda) that differ from the identity on only finitely many points. Such maps are called “almost everywhere the identity”. The authors begin by introducing the support of a map (\alpha), (\operatorname{supp}(\alpha)={x\in X\mid \alpha(x)\neq x}), which is always finite, and the “degree of deviation” (|\operatorname{supp}(\alpha)|). Using these notions they give a complete description of Green’s relations on (\mathscr{I}^{\infty}{\lambda}). Two elements are (\mathscr{J})‑equivalent precisely when they have the same deviation degree; consequently the (\mathscr{J})‑classes are indexed by the natural numbers. The (\mathscr{L})‑relation depends only on the size of the difference between the domains, while the (\mathscr{R})‑relation depends on the size of the difference between the ranges. Their intersection (\mathscr{H}) occurs only when both domain and range differences are equal, and (\mathscr{D}=\mathscr{J}). This shows that the Green structure collapses to a simple chain determined by a single integer invariant.

Next, the authors describe all two‑sided ideals. For each (k\in\mathbb{N}) they define \