Kuratowski Monoids on Posets

TL;DR

The paper extends Kuratowski-type phenomena from topological powersets to posets by studying the interplay of closure, interior, and pseudocomplement operators, and by embedding these into two completeness problems (CIP diagrams and their poset/locale variants). It proves that the interior–pseudocomplement monoid ${\mathbf M}$ has a finite, richly structured lattice of operators (at most $31$), and verifies a complete edge set via Bergman’s method; it also shows a corresponding pointfree (locale) diagram is complete, linking poset results to sublocales. The third major thread analyzes Chittenden semigroups ${\mathbf{Ch}}(m,n)$, proving a universal exponent $I(m,n)$ governing when $(st)^k=st$ and $(ts)^k=ts$ and enumerating global collapses (52 for $(3,3)$, 24 for $(2,3)$), with a general Finite theory for all $(m,n)$ driven by $d=\gcd(m-1,n-1)$. Collectively, these results deliver a comprehensive, finite classification of poset-based Kuratowski morphisms and their global collapses, connecting poset operator theory with locale/topos concepts and expanding the classical closure–complement paradigm to a broad, diagrammatically complete setting.

Abstract

Ciraulo recently showed that Kuratowski's closure-complement problem for arbitrary powersets of topological spaces extends constructively to the interior-pseudocomplement problem for arbitrary posets, using the closure-interior problem for posets (CIP) as a natural starting point. After a brief overview of CIP, we resolve two diagram-completeness problems left open by Ciraulo. Finally, we study operator semigroups arising from a little-known 1941 theorem of Chittenden, which generalizes CIP.

Figures (7)

• Figure 1: All $18$ Kuratowski monoids 1993_garel_olivier on $(L,\le)$, up to order duality.
• Figure 2: The Hasse diagram of $\mathbf{M}$2025_ciraulo. Dashed lines represent critical pairs.
• Figure 3: The closure--supplement monoid 2025_ciraulo on $(\textsf{S}(L),\subseteq)$. Dashed lines represent critical pairs.
• Figure 4: The set $\overset{\infty}{\underset{ n=2}{\bigcup}}\ \overset{n}{\underset{ m=2}{\bigcup}} \ {\overset{\infty}{\underset{ k=1}{{\bigcup}}}}\,P(m,n,k)$ where $P(m,n,k)$ is the partial order on $\{s,t\}^k$ in $\mathbf{Ch}(m,n)$.
• Figure 5: The Hasse diagram of $\mathbf{Ch}(m,n)$ for selected $(m,n)$.

Theorems & Definitions (40)

• Definition 1
• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Proposition 2
• Proposition 3
• proof
• Theorem 3
• proof