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Algebra Seminar

The Algebra Seminar provides graduate students, advanced undergraduate students, and faculty a forum to discuss and/or present Research, Problems, Applications, or Exposition in Algebra. For more information contact Arturo Magidin.

Spring 2017

During the Spring 2017 semester we will meet on Fridays from noon-12:50 in Maxim Doucet Hall room 214.

  • 27 January 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice?
    Arturo Magidin
     
    Abstract: (This is joint work with Martha Kilpack of Brigham Young University.)
    A lattice is a partially ordered set in which every pair of elements have a least upper bound and a greatest lower bound; it can also be viewed as an algebra with two binary operators satisfying certain identities. A typical example of a lattice is the lattice of all subgroups of a given group, ordered by inclusion (or more generally, of all substructures of a given structure).
    A closure operator on a partially ordered set P is a function f mapping P into P such that f satisfies three conditions:
    (i) f is increasing: x less than or equal to f(x) for all x in P;
    (ii) f is isotone: if x less than or equal to y, then f(x) less than or equal to f(y) for all x,y in P;
    (iii) f is idempotent: f(f(x)) = f(x).
    We can partially order all closure operators on P, by letting f less than or equal to g if and only if f(x) less than or equal to g(x) for all x in P. If P is a lattice, then this makes the set of all closure operators on P into a lattice.
    It is a theorem of Birkhoff that every complete lattice is the lattice of subalgebras of some (possibly infinitary algebra); and a theorem of Whitman that every lattice can be embedded as a sublattice of a subgroup lattice. This leads to the question of which lattices of closure operators are isomorphic to the subgroup lattice of a group.
    Previously, we had shown that if we look at the closure operators on a subgroup lattice Sub(G), then this resulting lattice, c.o.(Sub(G)) is itself a subgroup lattice if and only if G is cyclic of prime power order. We will extend the investigation first to the case of closure operators on an arbitrary finite lattice; and later, to the lattice of subgroups of an infinite group.
  • 3 February 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 2)
    Arturo Magidin
  • 10 February 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 3)
    Arturo Magidin
  • 17 February 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 4)
    Arturo Magidin
  • 24 February 2017
    If G is an infinite group, when is the lattice of algebraic closure operators on Sub(G) isomorphic to the subgroup lattice of a group K?
    Arturo Magidin
  • 3 March 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (last part)
    Arturo Magidin
  • 10 March 2017 (SPECIAL TIME 12:20)
    A description of indecomposable quasi-Frobenius rings
    Gary Birkenmeier
    Abstract In this talk, we characterize a nilary QF-ring R in terms of the essentiality of the ideals ReR where e is a primitive idempotent of R. Recall that a ring R is nilary if AB = 0 implies either A or B is nilpotent for all ideals A, B of R. Note that nilary rings are indecomposable rings. This is a preliminary report on joint research with Omar A. Al-Mallah andHafedh M. Al-Noghashi.
  • 24 March 2017
    A description of indecomposable quasi-Frobenius rings (part 2)
    Gary Birkenmeier
  • 31 March 2017
    A description of indecomposable quasi-Frobenius rings (part 3)
    Gary Birkenmeier
  • 7 April 2017
    A description of indecomposable quasi-Frobenius rings (part 4)
    Gary Birkenmeier