We introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the catego...

Main Theorem. For every odd prime power q = p, there is a symmetric design D with parameters (2q+1, q, q 2 −1 2 ) possessing an automorphism group A of order q · ( q−1 2 ) 2 · e · 2 which is isomorphic to the subdirect product of the affine semilinear group AΓL1(q) with itself with respect to a certain epimorphism ψ : AΓL1(q) → Z2 from AΓL1(q) into the cyclic group Z2, i.e. A ∼= AΓL1(q) sdp(ψ,ψ...

In this paper we introduce the concept of an (L; ')-representation of an algebra A which is a common generalization of subdirect, full subdirect and weak direct representation of A. Here we characterize such representations in terms of congruence relations.

Pałasińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in t...

The notion of subdirect irreducibility in the context of languages without equality, as presented by Elgueta, is extended in order to obtain subdirect representation theorems for abstract and reduced classes of structure systems. Structure systems serve as models of firstorder theories but, rather than having universal algebras as their algebraic reducts, they have algebraic systems in the sens...

If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex ∆(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M3 the 5-element nondistributive modular lattice...

In 2003, Di Nola, et.al. introduced the notion of MV -modules over a PMV algebra and A-ideals in MV -modules [5]. These are structures that naturally correspond to lu-modules over lu-rings [5]. Recall that an lu-ring is a pair (R,u), where (R, ⊕, ·, 0, ≤) is an l-ring and u is a strong unit of R (i.e, u is a strong unit of the underlying l-group) such that u · u ≤ u and l-ring is a structure (R...

It has been conjectured that the group algebra Q[G] of a discrete group G over the rational numbers Q is semi-simple. In this paper we consider irreducible representations of this algebra satisfying certain finiteness conditions and see just how much information these yield towards solving the above problem. We show in fact for a finitely generated group G, Q[G] has "enough" of these representa...

This paper shows that any compactly generated lattice is a subdirect product of subdirectly irreducible lattices which are complete and upper continuous. An example of a compactly generated lattice which cannot be subdirectly decomposed into subdirectly irreducible compactly generated lattices is given. In the case of an ideal lattice of a lattice L, the decomposition into subdirectly irreducib...

David Hilbert’s solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hilbert-NoetherBirkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere. The...