The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.

We prove that there exist profinite Heyting algebras that are not isomorphic to the profinite completion of any Heyting algebra. This resolves an open problem from 2009. More generally, we characterize those varieties of Heyting algebras in which profinite algebras are isomorphic to profinite completions. It turns out that there exists largest such. We give different characterizations of this variety and show that it is finitely axiomatizable and locally finite. From this it follows that it is decidable whether in a finitely axiomatizable variety of Heyting algebras all profinite members are profinite completions. In addition, we introduce and characterize representable varieties of Heyting algebras, thus drawing connection to the classical problem of representing posets as prime spectra.

We prove that if an n-element algebra generates the variety $$\mathcal {V}$$ which is actively structurally complete, then the cardinality of the carrier of each subdirectly irreducible algebra in $$\mathcal {V}$$ is at most $$n^{(n+1)\cdot n^{2\cdot n}}$$. As a consequence, with the use of known results, we show that there exist algorithms deciding whether a given finite algebra $$\mathbf {A}$$ generates the (actively) structurally complete variety $${\textsf {V}}(\mathbf {A})$$ in the cases when $${\textsf {V}}(\mathbf {A})$$ is congruence modular or $${\textsf {V}}(\mathbf {A})$$ is congruence meet-semidistributive or $$\mathbf {A}$$ is a semigroup.

The Blok-Esakia theorem states that there is an isomorphism from the lattice of intermediate logics onto the lattice of normal extensions of Grzegorczyk modal logic. The extension for multi-conclusion consequence relations was obtained by Emil Jeřábek as an application of canonical rules. We show that Jeřábek's result follows already from Blok's algebraic proof. We also prove that the properties of strong structural completeness and strong universal completeness are preserved and reflects by the aforementioned isomorphism. These properties coincide with structural completeness and universal completeness respectively for single-conclusion consequence relations and, in particular, for logics.

We provide simple algebraic proofs of two important facts, due to Zakharyaschev and Esakia, about Grzegorczyk algebras.

Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety V is standard if every Boolean topological algebra with the algebraic reduct in V is profinite. We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety V(A) generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson. We accomplish this by combining two results. The first one is Moore's theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether V(A) has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

We present a scheme for providing axiomatizations of universal classes. We use infinitary sentences there. New proofs of Birkhoff’s \(\mathsf {HSP}\)-theorem and Mal’cev’s \(\mathsf {SPP_U}\)-theorem are derived. In total, we present 75 facts of this sort.

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

Rab proteins, key players in vesicular transport in all eukaryotic cells, are post-translationally modified by lipid moieties. Two geranylgeranyl groups are attached to the Rab protein by the heterodimeric enzyme Rab geranylgeranyl transferase (RGT) αβ. Partial impairment in this enzyme activity in Arabidopsis, by disruption of the AtRGTB1 gene, is known to influence plant stature and disturb gravitropic and light responses. Here it is shown that mutations in each of the RGTB genes cause a tip growth defect, visible as root hair and pollen tube deformations. Moreover, FM 1-43 styryl dye endocytosis and recycling are affected in the mutant root hairs. Finally, it is demonstrated that the double mutant, with both AtRGTB genes disrupted, is non-viable due to absolute male sterility. Doubly mutated pollen is shrunken, has an abnormal exine structure, and shows strong disorganization of internal membranes, particularly of the endoplasmic reticulum system.

We study the following problem: Determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. As a consequence, we obtain a characterization of structurally complete discriminator varieties. An interesting corollary in logic follows: Let $L$ be a consistent propositional logic/deductive system in the language with formulas for verum, which is a theorem, and falsum, which is not a theorem. Assume also that $L$ has an adequate semantics given by a discriminator variety. Then $L$ is structurally complete if and only if it is maximal. All such logics/deductive systems are almost structurally complete.

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class 𝒞 of algebras let G(𝒞)={G(A)∣A∈𝒞}. Assume that 𝒞 is a class of semigroups possessing a nontrivial member with a neutral element and let ℋ be the universal Horn class generated by G(𝒞). We prove that the Boolean core of ℋ, i.e., the topological prevariety generated by finite members of ℋ equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ℋ. We derive analogous results when 𝒞 is a class of monoids or groups with a nontrivial member.

It is known that every Sikorski space with the countably generated differential structure is smoothly real-compact. It means that every homomorphism from its differential structure, which forms a ring of smooth real-valued functions into the ring of real numbers, is an evaluation. This result is sharp: there is a non-smoothly real-compact Sikorski space with the differential structure which is not countably generated. We provide an easy example demonstrating this. By modifying this example we are able to show a certain shortcoming of the generator embedding, comparing to the canonical embedding, for Sikorski spaces. Finally, we note that a homomorphism from the ring of smooth functions of a Sikorski space into the ring of real numbers is an evaluation if and only if it is continuous.

We note that a class of models is subdirect product closed if and only if it is definable by a class of L ∞∞-sentences of a special form.

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are \({\mathcal{Q}}\)-relation formulas for a protoalgebraic equality free quasivariety \({\mathcal{Q}}\) . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for \({\mathcal{Q}}\) when it has definable principal \({\mathcal{Q}}\)-subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties.

The graph of an algebra A is the relational structure G( A ) in which the relations are the graphs of the basic operations of A . Let denote by the class of all graphs of algebras from a class . We prove that if is a class of semigroups possessing a nontrivial member with a neutral element, then does not have finite quasi-equational basis. We deduce that, for a class of monoids or groups with a nontrivial member, also does not have finite quasi-equational basis.

The problem of embedding general algebras into modules is revisited. We provide a new method of embedding, based on JeZek's embedding into semimodules. We obtain several interesting consequences: a simpler syntactic characterization of quasi-affine algebras, a proof that quasi-affine algebras without nullary operations are actually quasi-linear, and several facts regarding the "abelian iff quasi-affine" problem.

. We describe the equational theory of the class of cancellative entropic algebras of a fixed type. We prove that a cancellative entropic algebra embeds into an entropic polyquasigroup, a natural generalization of a quasigroup. In fact our results are even more general and some corollaries hold also for non-entropic algebras. For instance an algebra with a binary cancellative term operation, which is a homomorphism, is quasi-affine. This gives a strengthening of K. Kearnes’ theorem. Our results generalize theorems obtained earlier by M. Sholander and by J. Ježek and T. Kepka in the case of groupoids.

An algebra is entropic if its basic operations are homomorphisms. The paper is focused on representations of such algebras. We prove the following theorem: An entropic algebra without constant basic operations which satisfies so called Szendrei identities and such that all its basic operations of arity at least two are surjective is a subreduct of a semimodule over a commutative semiring. Our theorem is a straightforward generalization of Ježek's and Kepka's theorem for groupoids. As a consequence we obtain that a mode (entropic and idempotent algebra) is a subreduct of a semimodule over a commutative semiring if and only if it satisfies Szendrei identities. This provides a complete solution to the problem in mode theory asking for a characterization of modes which are subreducts of semimodules over commutative semirings. In the second part of the paper we use our theorem to show that each entropic cancellative algebra is a subreduct of a module over a commutative ring. It extends a theorem of Romanowska and Smith about modes.

For quasivarieties of algebras, we consider the property of having definable relative principal subcongruences, a generalization of the concepts of definable relative principal congruences and definable principal subcongruences. We prove that a quasivariety of algebras with definable relative principal subcongruences has a finite quasiequational basis if and only if the class of its relative (finitely) subdirectly irreducible algebras is strictly elementary. Since a finitely generated relatively congruence-distributive quasivariety has definable relative principal subcongruences, we get a new proof of the result due to D. Pigozzi: a finitely generated relatively congruence-distributive quasivariety has a finite quasi-equational basis.