Let n ≥ 3. From the description of subdirectly irreducible complemented Arguesian lattices with four generators given by Herrmann, Ringel and Wille it follows that the subspace lattice of an n-dimensional vector space over a finite field is generated by four elements if and only if the field is a prime field. By exhibiting a 5-element generating set we prove that the subspace lattice of an n-di...

In the present paper we will ask for the lattice L(MVEx) of subvarieties of the variety defined by the set Ex(MV) of all externally compatible identities valid in the variety MV of all MV-algebras. In particular, we will find all subdirectly irreducible algebras from the classes in the lattice L(MVEx) and give syntactical and semantical characterization of the class of algebras defined by P -co...

Certainly, one of the most fundamental early results in universal algebra is Birkhoff’s 1944 decomposition theorem which states that every (universal) algebra d may be represented as a subdirect product of subdirectly irreducible algebras of the same type. It is clear that for d finite such decompositions may be obtained effectively, at least in principle. At first sight, therefore, one might b...

It is well known that there are five classes of sloops of cardinality 16 " SL(16)s" according to the number of sub-SL(8)s [4, 6]. In this article, we will show that there are exactly 8 classes of nonsimple sloops and 6 classes of simple sloops of cardinality 20 "SL(20)s". Based on the cardinality and the number of (normal) subsloops of SL(20), we will construct in section 3 all possible classes...

Let V be a variety whose class of finite members has a decidable first-order theory. We prove that each finite member A of V satisfies the (3,1) and (3,2) transfer principles, and that the minimal sets of prime quotients of type 2 or 3 in A must have empty tails. The first result has already been used by J. Jeong [9] in characterizing the finite subdirectly irreducible members of V with nonabel...

Basic Logic was introduced by Petr Hájek to provide a uni ed approach to fuzzy logics, and judging by its rapid adoption in the research community, it has enjoyed considerable success in this regard. One of the reasons is that while it is a very general logic, it has elegant semantics with respect to the real unit interval, which allow for practical applications and use of standard software too...

A commutative residuated lattice is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra A*). It proved here that epimorphisms surjective in variety K such algebras (with or without involution), provided each finitely subdirectly irreducible B has two properties: (1) generated by e, and (2) poset prime...

We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: (1) Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz’s prope...

In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively (finitely) subdirectly irreducible models. We identify two syntactical notions formulated in terms of (completely) intersection-prime theories that follow from finit...