Mini-course in Discrete Mathematics: "Relatively free Doppelsemigroups"

25.10. bis 27.10.2017  –  Campus Golm Workshop

A. V. Zhuchok

A. V. Zhuchok (Luhansk Taras Shevchenko National University, Starobilsk, Ukraine)

25.10.2017 14:15 – 15:45 in 2.09.1.10
26.10.2017 14:15 – 15:45 in 2.31.0.18
27.10.2017 14:15 – 15:45 in 2.09.0.14

Abstract. A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. We consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as interassociative semigroups, restrictive bisemigroups, and trioids.

Lecture 1. Examples and Independence of Axioms
We give an introduction to the topic: discuss the origin of doppelsemigroups and their connections with other algebraic structures. Numerous examples of doppelsemigroups and of strong doppelsemigroups are given. The independence of axioms of a doppelsemigroup is established.

Lecture 2. Doppelsemigroups
We construct a free product of doppelsemigroups and, as a consequence, obtain a free doppelsemigroup. We also construct a free commutative doppelsemigroup, a free n-nilpotent doppelsemigroup, a free n-dinilpotent doppelsemigroup, and a free left (right) n-dinilpotent doppelsemigroup. Moreover, the least commutative congruence, the least n-nilpotent congruence, the least n-dinilpotent congruence, and the least left (right) n-dinilpotent congruence on a free doppelsemigroup are characterized.

Lecture 3. Strong Doppelsemigroups
We construct a free strong doppelsemigroup and for this doppelsemigroup we give an isomorphic construction. We also construct a free n-dinilpotent strong doppelsemigroup, a free commutative strong doppelsemigroup, and a free n-nilpotent strong doppelsemigroup. Besides, the least n-dinilpotent congruence, the least commutative congruence, and the least n-nilpotent congruence on a free strong doppelsemigroup are presented. We also establish that the automorphism groups of the constructed free algebras are isomorphic to the symmetric group.

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