Categorical Abstract Algebraic Logic: More on Protoalgebraicity

Abstract

Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π-institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics.

Categorical abstract algebraic logic: The largest theory system included in a theory family

“…This characterization fills in a gap that was left open in the development of the theory of prealgebraicity and protoalgebraicity of [18]. As another important consequence of Theorem 3.2, it is shown that, in the context of N -protoalgebraic π-institutions, the N -Leibniz theory family T (N ) corresponding to a given theory family T , defined in a way analogous to the N -Leibniz theory system T N corresponding to a given theory system T in [20], is an N -Leibniz theory system. Therefore all N -Leibniz theory families are N -Leibniz theory systems and the theory of [20] covers these families in their full generality.…”

“…This result has two very important consequences for the theory of categorical abstract algebraic logic. On the one hand, it provides a characterization of the class of N -protoalgebraic π-institutions inside the class of N -prealgebraic π-institutions, as introduced in [18], and, on the other hand, it yields the interesting property that the N -Leibniz theory family T (N ) corresponding to a given theory family T of an N -protoalgebraic π-institution I is an N -Leibniz theory system, as introduced in [20].…”

“…This attempt was primarily based on a novel notion of a Leibniz operator for π-institutions, the N -Leibniz operator, first introduced in [12] and further elaborated on in [13,14,15,16,17,18,19,20]. Recall that, given a π-institution I = Sign, SEN, C , a theory family of I is a collection T = {T Σ } Σ∈|Sign| , such that T Σ ⊆ SEN(Σ) is a Σ-theory of I, for all Σ ∈ |Sign|.…”

“…In [20], the notion of a Leibniz theory of a sentential logic [7,8] was adapted in order to obtain the corresponding notion of an N -Leibniz theory system of a π-institution. Since in π-institutions the dichotomy between theory families and theory systems is ever present, it may be possible, to each given theory family T in an N -protoalgebraic π-institution, to associate either its N -Leibniz theory family or its N -Leibniz theory system by taking intersections of theory families or theory systems, respectively, having the same Leibniz N -congruence system.…”

Categorical abstract algebraic logic: The categorical Suszko operator

“…In a way very similar to the way an N -parameterized equivalence system behaves with respect to the N -Leibniz operator on π-institutions (see, e. g., [24,Proposition 4.1]), the analytical relation system E(T ) on SEN determined by E and T has the property that, if it is reflexive, then it contains the N -Suszko congruence system of T . P r o o f. In fact, suppose that Σ ∈ |Sign|, ϕ, ψ ∈ SEN(Σ) such that ϕ, ψ ∈ Θ N Σ (T ).…”