We show that the properties of [relative] semisimplicity and congruence 3-permutability of a [quasi]variety with equationally definable [relative] principal congruences (EDP[R]C) can be characterized syntactically. We prove that a quasivariety with EDPRC is relatively semisimple if and only if it satisfies a finite set of quasi-identities that is effectively constructible from any conjunction of equations defining relative principal congruences in the quasivariety. This in turn allows us to obtain an ‘axiomatization’ of relatively filtral quasivarieties. (...) We also show that a variety is 3-permutable and has EDPC if and only if there is a single pair of quaternary terms satisfying two simple equations, and whose equality defines principal congruences in the variety. Finally, we combine both results to obtain a neat characterization of semisimple, 3-permutable varieties with EDPC, which is applied to solve a problem posed by Blok and Pigozzi in the third paper of their series on varieties with EDPC. (shrink)

We prove that in a locally finite variety that has definable principal congruences , solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of (...) a model completion implies that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion. (shrink)

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. (shrink)

In this paper we pursue the study of the variety of m -generalized Łukasiewicz algebras of order n which was initiated in [1]. This variety contains the variety of Łukasiewicz algebras of order n . Given , we establish an isomorphism from its congruence lattice to the lattice of Stone filters of a certain Łukasiewicz algebra of order n and for each congruence on A we find a description via the corresponding Stone filter. We characterize the principal congruences (...) on A via Stone filters. In doing so, we obtain a polynomial equation which defines the principal congruences on the algebras of . After showing that for m > 1 and n > 2, the variety of Łukasiewicz algebras of order n is a proper subvariety of , we prove that is a finitely generated discriminator variety and point out some consequences of this strong property, one of which is congruence permutability. (shrink)

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown to (...) be equivalent to the finite identity basis problem for the finite members of a finitely based variety with definable principal congruences. (shrink)

This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic |$FL_{ew}$| (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. |$FL_{ew}$|-algebras) that includes both Heyting and Nelson algebras (...) and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics. (shrink)

We investigate expansions of Heyting algebras in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators. We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal (...) class='Hi'>congruences using a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations. (shrink)

We investigate expansions of Heyting algebras in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators. We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal (...) class='Hi'>congruences using a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations. (shrink)

In this paper, we show that all semisimple varieties of bounded weak-commutative residuated lattices with an S4-like modal operator are discriminator varieties. We also give a characterization of discriminator and EDPC varieties of bounded weak-commutative residuated lattices with an S4-like modal operator follows.

It is proved that the directly indecomposable algebras in a congruence modular equational class form a first-order class provided that fulfils some two natural assumptions.

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual (...) DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

School principals world-wide report high levels of strain and attrition resulting in a shortage of qualified principals. It is thus, crucial to identify psychosocial risk factors that reflect principals’ occupational wellbeing. For this purpose, we used the Copenhagen Psychosocial Questionnaire (COPSOQ-II), a widely used self-report measure covering multiple psychosocial factors identified by leading occupational stress theories. We evaluated the COPSOQ-II regarding factor structure and longitudinal, discriminant, and convergent validity using latent structural equation modeling in a large sample of Australian school (...) principals (N = 2,049). Results reveal that confirmatory factor analysis produced marginally acceptable model fit. A novel approach we call set exploratory structural equation modeling (ESEM-set), where cross-loadings were only allowed within a priori defined sets of factors, fit well, and was more parsimonious than a full ESEM. Further multitrait-multimethod models based on the set-ESEM confirm the importance of a principal’s psychosocial risk factors; Stressors and depression were related to demands and ill-being, while confidence and autonomy were related to wellbeing. We also show that working in the private sector was beneficial for showing a low psychosocial risk, while other demographics have little effects. Finally, we identify five latent risk profiles (high risk to no risk) of school principals based on all psychosocial factors. Overall the research presented here closes the theory application gap of a strong multi-dimensional measure of psychosocial risk-factors. (shrink)

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

Finitely algebraizable deductive systems were introduced by Blok and Pigozzi to capture the essential properties of those deductive systems that are very tightly connected to quasivarieties of universal algebras. They include the equivalential logics of Czelakowski. Based on Blok and Pigozzi’s work, Herrmann defined algebraizable deductive systems. These are the equivalential deductive systems that are also truth-equational, in the sense that the truth predicate of the class of their reduced matrix models is explicitly definable by some set of unary (...) equations. Raftery undertook the task of characterizing the property of truth-equationality for arbitrary deductive systems. In this paper, following Raftery, we extend the notion of truth-equationality for logics formalized as $\pi$-institutions and abstract several of the results that hold for deductive systems in this more general categorical context. (shrink)

In this paper we use Hobby's duality for semi-De Morgan algebras, to characterize those algebras having only principal congruences in the classes of semi-De Morgan algebras, demi-pseudocomplemented lattices and almost pseudocomplemented lattices. This work extends some of the results reached by Beazer in [3] and [4].

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

This paper presents the first purely algebraic characterization of classes of partial algebras definable by a set of strong equations. This result was posible due to new tools such as invariant congruences, i.e. a generalization of the notion of a fully invariant congruence, and extension of algebras, specific for strong equations.

The experimental setup of the self-referential quantum measurement, jovially known as the ‘quantum suicide’ or the ‘quantum Russian roulette’ is analyzed from the point of view of the Principal Principle of David Lewis. It is shown that the apparent violation of this principle – relating objective probabilities and subjective chance – in this type of thought experiment is just an illusion due to the usage of some terms and concepts ill-defined in the quantum context. We conclude that even in (...) the case that Everett’s (or some other ‘no-collapse’) theory is a correct description of reality, we can coherently believe in equating subjective credence with objective chance in quantum-mechanical experiments. This is in agreement with results of the research on personal identity in the quantum context by Parfit and Tappenden. (shrink)

The fact that it is possible to define three different material conditionals in orthomodular lattices suggests that there exist three different orthomodular logics whose conditionals are material conditionals and whose models are orthomodular lattices. The purpose of this paper is to provide equationally definable implication algebras for each of these material conditionals.

Let \({\mathcal {K}}\) be a quasivariety. We say that \({\mathcal {K}}\) is a term quasivariety if there exist an operation of arity zero _e_ and a family of binary terms \(\{t_i\}_{i\in I}\) such that for every \(A \in {\mathcal {K}}\), \(\theta \) a \({\mathcal {K}}\) -congruence of _A_ and \(a,b\in A\) the following condition is satisfied: \((a,b)\in \theta \) if and only if \((t_{i}(a,b),e) \in \theta \) for every \(i\in I\). In this paper we study term quasivarieties. For every \(A\in (...) {\mathcal {K}}\) and \(a,b\in A\) we present a description for the smallest \({\mathcal {K}}\) -congruence containing the pair (_a_, _b_). We apply this result in order to characterize \({\mathcal {K}}\) -compatible functions on _A_ (i.e., functions which preserve all the \({\mathcal {K}}\) -congruences of _A_) and we give two applications of this property: (1) we give necessary conditions on \({\mathcal {K}}\) for which for every \(A \in {\mathcal {K}}\) the \({\mathcal {K}}\) -compatible functions on _A_ coincides with a polynomial over finite subsets of _A_; (2) we give a method to build up \({\mathcal {K}}\) -compatible functions. (shrink)

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 ${\mathcal{Q}}$ Q -relation formulas for a protoalgebraic equality free quasivariety ${\mathcal{Q}}$ 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 ${\mathcal{Q}}$ Q when it has definable principal ${\mathcal{Q}}$ (...) 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. (shrink)

The IKt-algebras were introduced in the paper An algebraic axiomatization of the Ewald’s intuitionistic tense logic by the first and third author. In this paper, our main interest is to investigate the principal and Boolean congruences on IKt-algebras. In order to do this we take into account a topological duality for these algebras obtained in Figallo et al. :673–701, 2017). Furthermore, we characterize Boolean and principal IKt-congruences and we show that Boolean IKt-congruence are principal IKt- (...) class='Hi'>congruences. Also, bearing in mind the above results, we obtain that Boolean IKt-congruences are commutative, regular and uniform. Finally, we characterize the principal IKt-congruences in the case that the IKt-algebra is linear and complete whose prime filters are complete and also the case that it is linear and finite. This allowed us to establish that the intersection of two principal IKt-congruences on these algebras is a principal one and also to determine necessary and sufficient conditions so that a principal IKt-congruence is a Boolean one on theses algebras. (shrink)

An l-hemi-implicative semilattice is an algebra $$\mathbf {A} = $$ A= such that $$$$ is a semilattice with a greatest element 1 and satisfies: for every $$a,b,c\in A$$ a,b,c∈A, $$a\le b\rightarrow c$$ a≤b→c implies $$a\wedge b \le c$$ a∧b≤c and $$a\rightarrow a = 1$$ a→a=1. An l-hemi-implicative semilattice is commutative if if it satisfies that $$a\rightarrow b = b\rightarrow a$$ a→b=b→a for every $$a,b\in A$$ a,b∈A. It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide (...) a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an derived operation by $$a \sim b := \wedge $$ a∼b:=∧. Endowing $$$$ with the binary operation $$\sim $$ ∼ the algebra $$$$ results an l-hemi-implicative semilattice, which also satisfies the identity $$a \sim b = b \sim a$$ a∼b=b∼a. In this article, we characterize the commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element. Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices. (shrink)

An l-hemi-implicative semilattice is an algebra \\) such that \\) is a semilattice with a greatest element 1 and satisfies: for every \, \ implies \ and \. An l-hemi-implicative semilattice is commutative if if it satisfies that \ for every \. It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an (...) derived operation by \ \wedge \). Endowing \\) with the binary operation \ the algebra \\) results an l-hemi-implicative semilattice, which also satisfies the identity \. In this article, we characterize the commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element. Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices. (shrink)

We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω (Th) or countably compact regular sublogic ofL ∞ω (Th), properly extendingL ωω , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL (...) ∞ω (Th) in which Chang's quantifier or some cardinality quantifierQ α, with α≧1, is definable. (shrink)

\ of monadic m-generalized Łukasiewicz algebras of order n -algebras), namely a generalization of monadic n-valued Łukasiewicz algebras. In this article, we determine the congruences and we characterized the subdirectly irreducible \-algebras. From this last result we proved that \ is a discriminator variety and as a consequence we characterized the principal congruences. In the last part of this paper we find an immersion of these algebras in a functional algebra and we proved that in the finite (...) case they are isomorphic. This last result allows to show a new functional representation for monadic n-valued Łukasiewicz algebras. Finally, we define the notions of \-algebra of fractions and maximal algebra of fractions and we prove the existence of a maximal \-algebra of fractions for an \-algebra. (shrink)

After an introduction which demonstrates the failure of the equational analogue of Beth?s definability theorem, the first two sections of this paper are devoted to an elementary exposition of a proof that a functional constant is equationally definable in an equational theory iff every model of the set of those consequences of the theory that do not contain the functional constant is uniquely extendible to a model of the theory itself.Sections three, four and five are devoted to applications (...) and extensions of this result.Topics considered here include equational definability in first order logic, an extended notion of definability in equational logic and the synonymy of equational theories.The final two sections briefly review some of the history of equational logic. (shrink)

Teacher self-efficacy is one of the most critical factors influencing Students’ learning outcomes. Studies have shown that teacher-perceived principal leadership, teacher collaboration, and teaching experience are the critical factor that affects teacher self-efficacy. However, little is known about the mechanisms behind this relationship. This study examined whether teacher collaboration would mediate the relationship between teacher-perceived principal leadership and teacher self-efficacy, and the moderating role of teaching experience in the mediating process. With an analysis of a dataset from 14,121 (...) middle school teachers in China, this study first testified to the positive role that teacher-perceived principal leadership played in teacher self-efficacy. Furthermore, it revealed that teacher collaboration mediates this relationship and the mediated path was moderated by teaching experience. Finally, it also indicated that the threshold of teaching experience linking the teacher-perceived leadership with teacher self-efficacy was approximately in the third year, and their relationship was stronger when teaching experience was below the threshold. This study highlighted the mediating and moderating mechanisms linking the teacher-perceived principal leadership and teacher self-efficacy, which has important theoretical and practical implications for intervention and enhancement of teacher self-efficacy. (shrink)

In previous work, we have introduced and studied a lifting property in congruence–distributive universal algebras which we have defined based on the Boolean congruences of such algebras, and which we have called the Congruence Boolean Lifting Property. In a similar way, a lifting property based on factor congruences can be defined in congruence–distributive algebras; in this paper we introduce and study this property, which we have called the Factor Congruence Lifting Property. We also define the Boolean Lifting Property (...) in varieties with \ and \ having Boolean Factor Congruences and no skew congruences, and prove that it coincides to the Factor Congruence Lifting Property in the congruence–distributive case; we particularize this result to bounded distributive lattices and residuated lattices. (shrink)

We continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$ -equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$ -divisible. Here we introduce a new tool: a relation of congruence modulo an (...) ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_{\sim }$ -equivalent ultrafilters do not necessarily have the same residue modulo $m\in \mathbb {N}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we introduce a strengthening of $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ and show that it also behaves well with respect to the congruence relation. (shrink)