the aim of this paper is to extend results established by h. onoand t. kowalski regarding directly indecomposable commutative residuatedlattices to the non-commutative case. the main theorem states that a residuatedlattice a is directly indecomposable if and only if its boolean center b(a)is {0, 1}. we also prove that any linearly ordered residuated lattice and anylocal residuated lattice are d...

Let Var(Mplan) denote the variety generated by the class Mplan of planar modular lattices. In 1977, based on his structural investigations, R. Freese proved that Var(Mplan) has continuumly many subvarieties. The present paper provides a new approach to this result utilizing lattice identities. We also show that each subvariety of Var(Mplan) is generated by its planar (subdirectly irreducible) m...

A hoop is a naturally ordered pocrim (i.e., a partially ordered com-mutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its nite members.

Recently, the first two authors characterized in Di Nola and Dvurečenskij (2009) [1] subdirectly irreducible state-morphism MV-algebras. Unfortunately, the main theorem (Theorem 5.4(ii)) has a gap in the proof of Claim 10, as the example below shows. We now present a correct characterization and its correct proof. © 2010 Elsevier B.V. All rights reserved.

Lattice of subquasivarieties of variety generated by modular ortholattices MOn, n 2 ! and MO! is described. In [3] Igo sin has proved that any subquasivariety of the variety M! is a variety. M! denoted variety generated by modular lattice M!, where M! is the lattice of length two with ! atoms. For short proof of this fact see [2]. In this note we present an orthomodular counterpart of this resu...

A shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra. 2000 Mathematics subject classification: primary 03C05; secondary 08B05, 08B26.

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality...