The first fatal axiom for weakened sequential products on finite MV-effect algebras: Local obstruction, exact low-rank classification, and the rank-one boundary case

TL;DR

This paper investigates the structural limitations of sequential products on finite MV-effect algebras, revealing that the first fatal axiom appears at (S4) and providing a classification of operations on these algebras.

Contribution

It identifies the precise axiom (S4) as the first obstacle to defining sequential products on finite MV-effect algebras and classifies all such operations in specific cases.

Findings

Finite MV-effect algebras admit no (S1)-(S4) operation unless they are Boolean.

On the rank-two Boolean algebra, all (S1)-(S3) operations are classified with exactly 34 solutions.

The collapse at (S3) is a rank-one boundary phenomenon, with the threshold for nonexistence at (S4).

Abstract

We study total binary operations on effect algebras obtained by truncating the Gudder-Greechie axiom package for a sequential product. The point is not to reprove the known nonexistence of non-Boolean full sequential products on finite chains, but to determine, axiom by axiom, where finite MV-effect algebras first fail. We prove two structural facts valid on every effect algebra. First, the operation \sigma_E(a,b) = 0 if a=0, and b if a \neq 0, satisfies (S1)-(S3), so (S3) is never fatal by itself. Second, any operation satisfying (S1)-(S4) already has the right-unit property a \circ 1 = a, even without (S5). From this we derive a local obstruction theorem: if an effect algebra contains an atom of finite isotropic index at least 2, then it admits no (S1)-(S4) operation. Consequently, a finite MV-effect algebra admits such an operation if and only if it is Boolean. In this precise sense, (S4) is the first fatal axiom on finite MV-effect algebras. On the constructive side, let E_u = [0,u] \subseteq Z^r be the simplicial interval representation of a finite MV-effect algebra. We show that additive maps E_u \to E_v are exactly the restrictions of positive group homomorphisms Z^r \to Z^s, equivalently maps x \mapsto Mx given by nonnegative integer matrices with Mu \le v. This yields a complete classification of (S1)+(S2) operations by row-wise subunital matrices. We then solve the first genuinely higher-rank (S1)-(S3) problem: on the rank-two Boolean algebra B_2 = E_{(1,1)} \cong C_1^2, all such operations are classified and there are exactly 34. Thus the finite-chain collapse at (S3) is a rank-one boundary phenomenon, whereas on finite MV-effect algebras the sharp threshold for nonexistence occurs exactly at (S4).