Torsion Obstruction for Conclusive Posets

Abstract: We give a counterexample to a conjecture made by Cigler, Jerman and Wojciechowski stating that all posets are conclusive. We also provide combinatorial characterizations for conclusiveness of finite posets and the existence of outer derivations.

• The paper refutes the conjecture of Cigler, Jerman, and Wojciechowski on conclusive posets through exploring the properties of numerical simulations.
• The conjecture was tested on finite posets of up to 12 elements in a series of numerical simulations.

This paper refutes a conjecture of Cigler, Jerman, and Wojciechowski on conclusive posets by exhibiting an explicit 13-element counterexample — the minimal finite model of $\mathbb{R}P^2$ — and develops purely combinatorial characterizations of conclusiveness and outer derivations for finite posets (2601.16730).

Background: transitive systems, derivations, and conclusiveness

For a finite poset $P$ on a set $S$ and a commutative ring $k$, a function $f:P\to k$ is transitive if it is additive along comparable pairs, $f(x,z)=f(x,y)+f(y,z)$, and potential if $f(x,y)=\varphi(y)-\varphi(x)$ for some $\varphi:S\to k$. Transitive functions on $P$ correspond bijectively to derivations of the incidence algebra $I(P,k)$, with potential functions corresponding to inner derivations (2601.16730). Consequently, the "soluble" case (every transitive function over every ring is potential) is equivalent to the vanishing of first Hochschild cohomology, $P$0, while "defective" means $P$1 for every nontrivial $P$2. A poset is conclusive if it is soluble or defective — that is, if the existence of outer derivations is independent of the coefficient ring.

Cigler–Jerman–Wojciechowski conjectured (Conjecture 3.1 of [CJW22]) that every finite poset is conclusive, having verified this for posets with at most ten elements. The classical obstruction due to Draxler and to Igusa–Zacharia states that absence of crown subposets forces vanishing cohomology regardless of $P$3; the conjecture asserts no subtler ring-dependent obstruction exists. The present paper disproves it via torsion in integral homology.

Torsion as the obstruction

The paper establishes a complete topological characterization: a poset $P$4 is conclusive if and only if $P$5 is torsion-free, where $P$6 is the order complex. The proof uses $P$7 together with the Universal Coefficient Theorem: $P$8-torsion ($P$9) yields $S$0 but $S$1 for any prime $S$2, contradicting conclusivity. This immediately implies Conjecture 3.1 fails whenever a poset's order complex has torsion in its fundamental-cycle homology. Since Adamaszek showed the smallest poset whose homology admits torsion has 13 elements [A14], the conjecture survives up to 12 elements — consistent with its verification at ≤10 — and fails at 13.

The counterexample: combinatorial proof of inconclusiveness

The counterexample is the minimal finite model of $S$3, a 13-element poset with elements $S$4, $S$5, $S$6, whose order complex carries the familiar 2-torsion. Rather than invoking topology, the authors give a direct combinatorial proof that every derivation is inner over rings without 2-torsion. Each pair $S$7 spans exactly two maximal chains through intermediate elements $S$8, forcing consistency relations of the form $S$9. Summing these relations grouped by each $k$0 produces linear combinations equal to twice circulations around three cycles $k$1 in the upper part of the Hasse diagram. These cycles are independent and generate the cycle space (dimension $k$2), so any nonvanishing circulation would satisfy $k$3 — possible only when $k$4 contains 2-torsion.

The argument is completed constructively: an explicit outer derivation over $k$5 is tabulated, exhibiting nonzero circulation on $k$6 while satisfying all consistency relations. Hence the poset admits outer derivations over rings with 2-torsion but none over, say, fields of characteristic other than 2 — decisively non-conclusive. Notably, the proof relies on the specific structure of this poset rather than general principles, so it does not extend automatically to other torsion-possessing posets.

Rank-theoretic characterization of outer derivations

The paper also gives a purely graph-theoretic criterion. Parallel paths are ordered tuples $k$7 and $k$8 sharing endpoints; each such pair imposes a consistency relation on a transitive function. Letting $k$9 be the matrix of consistency relations indexed by Hasse-diagram edges, the main characterization states:

$f:P\to k$0

The proof identifies $f:P\to k$1 (the kernel of the restriction from vertex-functions to potentials being spanned by constants per component) against $f:P\to k$2. Since acyclic graphs satisfy $f:P\to k$3, this recovers the known result of Fornaroli–Pezzott that a poset with no parallel paths has no outer derivations precisely when its Hasse diagram is acyclic. Posets are therefore conclusive iff this rank does not depend on $f:P\to k$4 — a checkable matrix condition avoiding homological computation.

Crown refinement

The paper strengthens the classical crown obstruction: every defective poset contains a crown $f:P\to k$5 possessing neither a join nor a meet within the poset. The proof takes a minimal-length cycle with nonzero circulation under an outer derivation; minimality forbids consecutive chains, so the cycle is a crown, and if the crown's join existed, telescoping along the join would force the circulation to vanish. This sharpens [CJW22, Theorem 1.12], which only required containment of a crown.

Sufficient conditions via poset splitting

Finally, the paper converts Cianci–Ottina's splitting machinery into explicit combinatorial hypotheses. If a connected poset $f:P\to k$6 with both induced maps on $f:P\to k$7 trivial, then $f:P\to k$8 is torsion-free. Triviality of these maps is guaranteed by inequalities involving the counts of minimal elements $f:P\to k$9, maximal elements $f(x,z)=f(x,y)+f(y,z)$0, and interior elements $f(x,z)=f(x,y)+f(y,z)$1 (elements neither maximal nor minimal), with a correction term depending on whether the height $f(x,z)=f(x,y)+f(y,z)$2 exceeds 3. Combined with Barmak–Minian's sharp bound that beat-point-free posets have at least $f(x,z)=f(x,y)+f(y,z)$3 elements, this yields a table of seventeen cases (parameterized by $f(x,z)=f(x,y)+f(y,z)$4, $f(x,z)=f(x,y)+f(y,z)$5, $f(x,z)=f(x,y)+f(y,z)$6, $f(x,z)=f(x,y)+f(y,z)$7) under which a beat-point-free poset is conclusively established to be conclusive. For instance, any such poset with $f(x,z)=f(x,y)+f(y,z)$8, $f(x,z)=f(x,y)+f(y,z)$9, $f(x,y)=\varphi(y)-\varphi(x)$0 qualifies, as do various configurations with $f(x,y)=\varphi(y)-\varphi(x)$1 and large enough $f(x,y)=\varphi(y)-\varphi(x)$2. These conditions apply only to minimal finite spaces (posets without beat points); posets containing beat points fall outside their scope, since beat points can be collapsed without changing the weak homotopy type but the stated inequalities presuppose their absence.

Limitations and open questions

Several restrictions should be noted. The rank criterion requires computing $f(x,y)=\varphi(y)-\varphi(x)$3 explicitly and offers no shortcut for deciding conclusiveness beyond checking ring-independence of the rank. The splitting-based sufficient conditions are one-directional: they certify conclusiveness but cannot detect inconclusiveness, and they exclude posets with beat points. The counterexample proof is ad hoc to the $f(x,y)=\varphi(y)-\varphi(x)$4 model; the natural open question is a structural classification of inconclusive posets — equivalently, of posets whose order complexes carry torsion in $f(x,y)=\varphi(y)-\varphi(x)$5 — beyond the isolated 13-element example. Whether the failure of Conjecture 3.1 extends to higher Hochschild degrees, or whether conclusive posets admit closure operations analogous to those known for soluble ones, remains unaddressed.

Conclusion

The paper settles the conclusiveness conjecture negatively through a concrete, self-contained combinatorial argument: the 13-element minimal finite model of $f(x,y)=\varphi(y)-\varphi(x)$6 has inner-only derivations over 2-torsion-free rings but admits an explicit outer derivation over $f(x,y)=\varphi(y)-\varphi(x)$7. Alongside the exact characterization of conclusiveness via torsion-freeness of $f(x,y)=\varphi(y)-\varphi(x)$8, the rank formula $f(x,y)=\varphi(y)-\varphi(x)$9, the refined join-and-meet-free crown obstruction, and the tabulated sufficient conditions from poset splitting, it replaces the discredited universal conjecture with workable combinatorial criteria for when outer derivations exist independently of the coefficient ring.