### Algebraic properties of pre-logics

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The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra $H$ considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice $DedH$ of all deductive systems on $H$ and every maximal deductive system is prime. Complements and relative complements of $DedH$ are characterized as the so called annihilators in $H$.

An algebra $\mathcal{A}=(A,F)$ is subregular alias regular with respect to a unary term function $g$ if for each $\Theta ,\Phi \in \text{Con}\phantom{\rule{0.166667em}{0ex}}\mathcal{A}$ we have $\Theta =\Phi $ whenever ${\left[g\left(a\right)\right]}_{\Theta}={\left[g\left(a\right)\right]}_{\Phi}$ for each $a\in A$. We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset $C\subseteq A$ is a class of some congruence on $\Theta $ containing $g\left(a\right)$ if and only if $C$ is this generalized deductive system. This method is efficient (needs a finite number of steps).

In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal...

The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability...

The polyadic algebras that arise from the algebraization of the first-order extensions of a SIC are characterized and a representation theorem is proved. Standard implicational calculi (SIC)'s were considered by H. Rasiowa [19] and include classical and intuitionistic logic and their various weakenings and fragments, the many-valued logics of Post and Łukasiewicz, modal logics that admit the rule of necessitation, BCK logic, etc.