# Spring 2013 Seminars

**Time:** Tuesday, 23 April 2013, 4:10 PM

**Place: **SC 1312

**Speaker: **Alexander Wires

**Title:** A Disjunction for Varieties with a Weak Difference Term

**Abstract:** We provide a characterization for varieties with a weak difference term similar to the disjunction given for congruence meet-semidistributive varietes established by Ross Willard in “A finite basis theorem for residually finite, congruence meet-semidistributive varieties”(2000). As an application we give simplified proofs of two well-known results concerning the closure of digraphs under Taylor operations.

**Time: **Tuesday, 16 April 2013, 4:10 PM

**Place:** SC 1312

**Speaker: **Constantine Tsinakis

**Title:** The Finite Embeddability Property for Algebras of Logic II

**Abstract: ** This is a continuation of the previous talk on the finite embeddability property for algebras of logic. The discussion is based on a streamlined treatment of the theory of join-extensions for these algebras.

**Time: **Tuesday, 9 April 2013, 4:10 PM

**Place: **SC 1312

**Speaker:** Constantine Tsinakis

**Title:** The Finite Embeddability Property for Algebras of Logic

**Abstract:** The focus of this talk is the finite embeddability property for algebras of logic. The discussion is based on a streamlined treatment of the theory of join-extensions for these algebras.

**Time:** Tuesday, 2 April 2013, 4:10 PM

**Place:** SC 1312

**Speaker: **Ralph McKenzie

Title: Robust Maltsev conditions and Abelian algebras in varieties (II)

**Abstract:** Among others, I’ll be proving Keith Kearne’s result that Abelian algebras in a variety

having a Taylor term are quasi-affine (almost modules), and his characterization of the weakest

Maltsev condition for a variety that implies all Abelian algebras are affine (modules).

**Time:** Wednesday, 27 March 2013, 4:10 PM

**Place:** SC1210

**Speaker:** Keith Kearnes

**Title: **Growth Rate of Finite Algebras(Solvable)

**Abstract:**

**Time:** Tuesday, 26 March 2013, 4:10 PM

**Place: **SC 1312

**Speaker:** Russell Miller

**Title: **Independent sets in computable free groups and fields

**Abstract:** We consider maximal independent sets within various sorts of groups and fields freely generated by countably many generators. The simplest example is the free divisible abelian group, which is just an infinite-dimensional rational vector space. As one moves up to free abelian groups, free groups, and free fields; (i.e. purely transcendental field extensions), maximal independent sets and independent generating sets both become more complicated, from the point of view of computable model theory, but sometimes in unpredictable ways, and certain questions remain open. We present the topic partly for its own sake, but also with the intention of introducing the techniques of computable model theory and illustrating some of its possible uses for an audience to which it may be unfamiliar.

This is joint work with Charles McCoy.

**Time: **Tuesday, 19 March 2013, 11 AM

**Place:** SC 1431

**Speaker:** Williams Young

**Title:** An Investigation of Residuated Lattices with a Modal Operator

**Abstract:** Residuated lattices, which generalize Boolean algebras and lattice-ordered groups, have been useful in the study of algebraic logic, particularly as an algebraic semantics for substructural logics. By equipping a residuated lattice with a modal operator (either a nucleus or a conucleus) and then considering the image under this operator, a new class of residuated lattices results. For example, Heyting algebras arise as the conuclear images of Boolean algebras; commutative, cancellative residuated lattices as the conuclear images of Abelian lattice-ordered groups; and integral GMV-algebras as the nuclear images of negative cones of lattice-ordered groups.

In all three of the aforementioned cases, there is a categorical equivalence when you restrict to a certain subcategory of the class of residuated lattices with modal operators. Also, there is a strong connection between the subvariety lattice of the class of residuated lattices with modal operators and the corresponding class of their images under this operator. In particular, we show an even stronger connection in the case of negatively-pointed Abelian lattice-ordered groups (which can be seen as a residuated lattice with the nucleus that forms the interval from the designated negative element to the identity) and their images (which are MV-algebras). Namely, the subvariety lattice of negatively-pointed Abelian lattice-ordered groups (without the trivial variety) is isomorphic to the subvariety lattice of MV-algebras.

**Time:** Tuesday, 12 March 2013, 4:10 PM

**Place:** SC 1312

**Speaker: **Ralph McKenzie

**Title:** Robust Maltsev conditions and Abelian algebras in varieties, results old and new

**Abstract:** Among others, I’ll be proving Keith Kearne’s result that Abelian algebras in a variety

having a Taylor term are quasi-affine (almost modules), and his characterization of the weakest

Maltsev condition for a variety that implies all Abelian algebras are affine (modules).

**Time: **Tuesday, 26 February 2013, 4:10 PM

**Place: **SC 1312

**Speaker: **Michael Botur(Palacký University Olomouc)

**Title:** An elementary proof of the completeness of the Lukasiewicz axioms

**Abstract:** The main aim of talk is twofold. Firstly, to present an elementary method based on Farkas’ lemma for rationals on how to embed any finite partial subalgebra of a linearly ordered MV-algebra into $\mathbb Q\cap [0,1]$ and then to establish a new elementary proof of the completeness of the Lukasiewicz axioms. Secondly, to present a direct proof of Di Nola’s Representation Theorem for MV-algebras and to extend his results to the restriction of the standard MV-algebra on rational numbers.

**Time:** Tuesday, 19 February 2013, 4:10 PM

**Place:** SC 1312

**Speaker:** Jan Kuhr(Palacký University Olomouc)

**Title:** Divisible Psuedo-BCK Algebras

**Abstract:** A porim is called divisible if it is naturally ordered, in the sense that a b i

a = x b = b y for some x; y. Divisible porims are also known as pseudo-hoops,

and divisible integral residuated lattices as integral GBL-algebras. We focus on

divisibility in the setting of pseudo-BCK-algebras (or biresiduation algebras) that

are the residuation subreducts of porims. We attempt to generalize some structural

results proved by Blok and Ferreirim for hoops, and by Jipsen and Montagna for

integral GBL-algebras.

Since a porim is divisible i it satises the identities (xny)n(xnz) (ynx)n(ynz)

and (z=x)=(y=x) (z=y)=(x=y), it seems natural to call pseudo-BCK-algebras sat-

isfying these two identities divisible. Further, by a normal pseudo-BCK-algebra we

mean a pseudo-BCK-algebra A = hA; n; =; 1i in which the 1-classes of the relative

congruences can be characterized as the subsets K A satisfying: (i) 1 2 K,

(ii) for all a; b 2 A, if a; anb 2 K, then b 2 K, and (iii) for all a; b 2 A, anb 2 K i

b=a 2 K.

The main result for normal divisible pseudo-BCK-algebras is the following: If A

is a non-trivial subdirectly irreducible normal divisible pseudo-BCK-algebra, then

A is the ordinal sum BC, where C is a non-trivial subdirectly irreducible linearly

ordered cone algebra in the sense of Bosbach.

We also generalize the concept of n-potent porims and prove that every n-potent

divisible pseudo-BCK-algebra is a BCK-algebra.

**Time: **Tuesday, 12 February 2013, 4:10 PM

**Place: **SC 1312

**Speaker: **Chris Conidis

**Title: **The Reverse Mathematics of Prime Ideals in Commutative Rings

**Abstract:** We will show that the reverse mathematical strength of the statement “every commutative ring with identity has a prime ideal” is equivalent to WKL (Weak K\”onig’s Lemma) over RCA (Recursive Comprehension Axiom).

**Time: **Tuesday, 5 February 2013, 4:10 PM

**Place:** SC 1312

**Speaker: **Rebecca Steiner

**Title:** Is It Harder to Factor a Polynomial or to Find a Root?, Part II

**Abstract:** For a computable algebraic field F, the splitting set S_F of F is the set of polynomials with coefficients in F which factor over F, and the root set R_F of F is the set of polynomials with coefficients in F which have a root in F. In the first part of this talk, on October 2, 2012, we showed that under the bounded Turing (bT) reducibility, determining whether a polynomial has a root in a field F is more difficult than determining whether it factors over F, i.e. S_F is always bT-reducible to R_F, but there are fields F where R_F is not bT-reducible to S_F. In the second part, we will define a Rabin embedding g of a field into its algebraic closure, and for a computable algebraic field F, we compare the relative complexities of R_F, S_F, and g(F) under m-reducibility and under bT-reducibility.

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