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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2604.03324 |
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| _version_ | 1866911567000371200 |
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| author | Higuchi, Joaquim Reizi |
| author_facet | Higuchi, Joaquim Reizi |
| contents | 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 σ_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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03324 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 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 Higuchi, Joaquim Reizi Logic Logic in Computer Science Rings and Algebras 06D35, 81P10 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 σ_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). |
| title | 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 |
| topic | Logic Logic in Computer Science Rings and Algebras 06D35, 81P10 |
| url | https://arxiv.org/abs/2604.03324 |