Uniform interpolation and coherence

A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive interpolation: that is, any compact congruence on a finitely generated free algebra of V restricted to a free algebra over a subset of the generators is again compact. A general criterion is obtained for establishing failures of coherence, and hence also of uniform deductive interpolation. This criterion is then used in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (in particular, algebras of modal logic K and its standard non-transitive extensions), double-Heyting algebras, residuated lattices, and lattices

Locally Integral Involutive PO-Semigroups

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf A = (A,\le, \cdot, \sim,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\mathbf A_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $\Phi = \{\varphi_{pq}\: \mathbf A_p\to \mathbf A_q : p\le q\}$, indexed over the positive cone $(A^+,\le)$, so that the structure of $\mathbf A$ can be recovered as a glueing $\int_\Phi \mathbf A_p$ of its integral components along $\Phi$. Reciprocally, we give necessary and sufficient conditions so that the P{\l}onka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$, indexed over a join-semilattice $(D,\lor)$ along a family of monoid homomorphisms $\Phi$ is an ipo-semigroup

Order, algebra, and structure: lattice-ordered groups and beyond

This thesis describes and examines some remarkable relationships existing between seemingly quite different properties (algebraic, order-theoretic, and structural) of ordered groups. On the one hand, it revisits the foundational aspects of the structure theory of lattice-ordered groups, contributing a novel systematization of its relationship with the theory of orderable groups. One of the main contributions in this direction is a connection between validity in varieties of lattice-ordered groups, and orders on groups; a framework is also provided that allows for a systematic account of the relationship between orders and preorders on groups, and the structure theory of lattice-ordered groups. On the other hand, it branches off in new directions, probing the frontiers of several different areas of current research. More specifically, one of the main goals of this thesis is to suitably extend results that are proper to the theory of lattice-ordered groups to the realm of more general, related algebraic structures; namely, distributive lattice-ordered monoids and residuated lattices. The theory of lattice-ordered groups provides themain source of inspiration for this thesis’ contributions on these topics