Introduction: This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings—the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning. D. G. Johnson has shown [9] that not every /-ring is unitable, i.e. e...

The main goal in this paper is to use a dual equivalence in automata theory started in [RBBCL13] and developed in [BBCLR14] to prove a general version of the Eilenberg-type theorem presented in [BBPSE12]. Our principal results confirm the existence of a bijective correspondence between formations of (non-necessarily finite) monoids, that is, classes of monoids closed under taking epimorphic ima...

In 1998, Hájek established a representation theorem of BL-algebrs as all subdirect products of linear BL-algebrs. We establish a similar result for the much wider class of prelinear residuated algebras, in which neither the lattice structure nor the divisibility of the monoid operation is assumed. We show, in the case of prelinear residuated lattices, that this order embedding becomes a lattice...

All groups considered in this paper are tacitly assumed to be finite. Recall that a formation 5 is a class of groups which is closed under taking homomorphic images and subdirect products. It is known that, if 5 is a formation, each group G has a smallest normal subgroup whose factor group is in 5Such subgroup is called the ^-residual of G and it is denoted by G. A formation # is said to be sat...

The scaffolding of a complete lattice L of finite length was introduced by Rudolf Wille in 1976 as a relative subsemilattice of L that can be constructed using subdirect decomposition. The lattice is uniquely defined by its scaffolding and can be reconstructed from it. Using bonds, we demonstrate how the scaffolding can be constructed from a given formal context and thereby extend the notion of...

CONTENTS This article proposes to represent subgroups of finitely presented 1 . Creation of Quotient Representations and Easy Calculations g r o u p s b y t h e i r i m a g e i n a quo t ient. It gives algorithms for basic 2. Subdirect Products operations in this representation and investigates how iteration 3. Iterated Quotient Representations of this approach can be used to extend known quoti...

This paper concerns residuated lattice-ordered idempotent commutative monoids that are subdirect products of chains. An algebra of this kind is a generalized Sugihara monoid (GSM) if it is generated by the lower bounds of the monoid identity; it is a Sugihara monoid if it has a compatible involution ¬. Our main theorem establishes a category equivalence between GSMs and relative Stone algebras ...