Given a semigroup S and elements s,t in S, write s~t if s=pr and t=rp, for some p,r in S (with 1 adjoined). This relation, and its transitive closure, both known as "primary conjugacy", has been extensively used and studied in connection with many algebraic objects, including groups, rings, and C*-algebras. It is the standard tool for either measuring or forcing commutativity. We discuss a natural generalization, defined by s~t whenever s=p_{1}...p_{n} and t=p_{f(1)}...p_{f(n)}, for some p_{1},...,p_{n} in S (with 1 adjoined) and permutation f of {1,...,n}, together with its transitive closure, which we call the "permutation" relation. The permutation relation is actually the congruence generated by the primary conjugacy, and is the least commutative congruence on any semigroup. We explore general properties of the permutation relation, discuss it in the context of groups and rings, compare it to various known semigroup conjugacy relations, and describe its equivalence classes in different semigroups.
Conjugacy and least commutative congruences in semigroups Sponsored by the Meyer Fund
Oct. 28, 2025 2:30pm (MATH 2…
Grad Algebra/Logic
Charlotte Aten (CU Boulder)
X
I will state the fundamental theorem of the algebraic approach to promise constraint satisfaction and present a family of examples where we can explicitly see the connection between gadgets, nerves, and minion homomorphisms.
Examples for the Fundamental Theorem of Algebraic PCSP
Oct. 28, 2025 3:30pm (MATH 3…
Topology
Alex LaJeunesse
X
Machinery from homotopy theory allows us to define a Brauer space for any E_3-ring R whose homotopy groups carry important arithmetic information about R. If R is equivalent to a discrete commutative ring, for example, these homotopy groups agree with the classical (derived) unit, Picard, and Brauer groups. I will describe how one constructs this Brauer space and discuss the problem of "strictification" for Brauer classes.
Conjugacy and least commutative congruences in semigroups