An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Locally Integral Involutive PO-Semigroups

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf A = (A,\le, \cdot, \sim,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\mathbf A_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $\Phi = \{\varphi_{pq}\: \mathbf A_p\to \mathbf A_q : p\le q\}$, indexed over the positive cone $(A^+,\le)$, so that the structure of $\mathbf A$ can be recovered as a glueing $\int_\Phi \mathbf A_p$ of its integral components along $\Phi$. Reciprocally, we give necessary and sufficient conditions so that the P{\l}onka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$, indexed over a join-semilattice $(D,\lor)$ along a family of monoid homomorphisms $\Phi$ is an ipo-semigroup

The Structure of Locally Integral Involutive Po-monoids and Semirings

We show that every locally integral involutive partially ordered monoid (ipo-monoid) A = (A,⩽, ·, 1,∼,−), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {Ap : p ∈ A+} of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are semirings. Moreover, we show that there is a family of monoid homomorphisms Φ = {φpq : Ap → Aq : p ⩽ q}, indexed on the positive cone (A+,⩽), so that the structure of A can be recovered as a glueing R ΦAp of its integral components along Φ. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids {Ap : p ∈ D}, indexed on a lower-bounded join-semilattice (D,∨, 1), along a family of monoid homomorphisms Φ is an ipo-monoid

Structure Theorems for Idempotent Residuated Lattices

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property

Structure theorems for idempotent residuated lattices

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property

Modules over quantaloids: applications to the isomorphism problem in algebraic logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions

Modules over Quantaloids: Applications to the Isomorphism Problem in Algebraic Logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions