One of the central themes in classifying varieties V which
have the property that the satisfiability problem for finite members
of V is decidable, is that local structural properties can have
outsized global consequences. In this talk we will see the proof that
if a finite algebra in V has a type-2 or type-3 minimal set with
nonempty tail, then it is undecidable whether a first-order sentence
in the language of V is satisfiable in a finite member of V; indeed,
that Th(V) is recursively inseparable from the sentences refuted in
finite members of V.

This talk will introduce the concept of an algebra induced by a unary term. If A is an algebra in the variety V and \tau is a unary term in the language of V, the induced algebra A_{\tau} is the image of A under \tau, together with the induced operations defined by taking \tau of the original operations. It will be shown that this map is in fact a functor \Lambda from V to W, the class of subalgebras of the induced algebras.The main result of the talk is that the free algebra in W is a subalgebra of \Lamdba of the free algebra in V. After proving this, we will go through two specific examples of the construction to illustrate its usefulness.

We will continue the proof of the following recent result of M. Maroti and L. Zadori: A finite reflexive digraph admits Gumm operations iff it admits a near-unanimity operation.

Joint Meeting with Group Theory and Topology Seminar

Let R be a commutative ring with identity. Recall from basic graduate algebra that:

1. R is Noetherian if it satisfies the ascending chain condition on its ideals;
2. R is Artinian if it satisfies the descending chain condition on its ideals; and
3. R is of finite length if there is a uniform bound on the length of any (strictly increasing/decreasing) chain of ideals in R.

It is well-known that 2. implies 1., but the proofs given in most standard algebra texts prove this by showing the stronger statement that 2. implies 3. This begs the question: “Can one prove that 2. implies 1. without showing that 2. implies 3? We will show that this is indeed the case by showing that, in the context of reverse mathematics, the former (weaker) statement is equivalent to weak Konig’s lemma, while the latter (stronger) statement is equivalent to arithmetic comprehension in the context of $omega$-models. Another way to view our main result is that it constructs an $\omega$-model of RCA (recursive comprehension axiom) in which 2. implies 1., but 2. does not imply 3.

We will study and prove the following recent result of M. Maroti and L. Zadori: A finite reflexive digraph admits Gumm operations iff it admits a near-unanimity operation. The result is important for the algebraic study of Constraint Satisfaction Problems and for the recently studied phenomenon of Malcev collapse for finitely related algebras.

Sugihara monoids (idempotent involutive semilinear commutative residuated
lattices) provide an equivalent algebraic semantics for the well known
“R-Mingle with unit” logic RMe. This class of algebras forms a locally finite
variety SM generated by a single algebra on the integers with zero removed.
In this talk I will first give a complete description of the subvariety lattice of SM,
and then explain how to show that exactly eight of these subvarieties have the
amalgamation property. This corresponds to the fact that RMe has exactly
eight axiomatic extensions with the deductive interpolation property or,
equivalently (perhaps surprisingly), the Craig interpolation property.
(Joint work with Enrico Marchioni)