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Márki László (Rényi Intézet): Kommutatív hányadosfélcsoportok (Commutative orders in semigroups)

 iCal fájl letöltése
Szerda, 21. Október 2015, 10:00 - 12:00
Abstract: We consider commutative orders, that is, commutative semigroups having a semigroup of quotients in a local sense defined as follows. An element a ∈ S is square cancellable if for all x, y ∈ S¹ we have that xa² = ya² implies xa = ya and also a²x = a²y implies ax =ay. It is clear that being square-cancellable is a necessary condition for an element to lie in a subgroup of an oversemigroup. In a commutative semigroup S, the square-cancellable elements constitute a subsemigroup S(S). Let S be a subsemigroup of a semigroup Q. Then S is a left order in Q and Q is a semigroup of left quotients of S if every q ∈ Q can be written as q = a♯b where a ∈ S(S), b ∈ S and a♯ is the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Right orders and semigroups of right quotients are defined dually. If S is both a left order and a right order in Q, then S is an order in Q and Q is a semigroup of quotients of S. We remark that if a commutative semigroup is a left order in Q, then Q is commutative so that S is an order in Q. A given commutative order S may have more than one semigroup of quotients. The semigroups of quotients of S are pre-ordered by the relation Q ≥ P if and only if there exists an onto homomorphism φ : Q → P which restricts to the identity on S. Such a φ is referred to as an S-homomorphism; the classes of the associated equivalence relation are the S-isomorphism classes of orders, giving us a partially ordered set Q(S). In the best case, Q(S) contains maximum and minimum elements. In a commutative order S, S(S) is also an order and has a maximum semigroup of quotients R, which is a Clifford semigroup. We investigate how much of the relation between S(S) and its semigroups of quotients can be lifted to S and its semigroups of quotients.
Hely : Bolyai Intézet, I. emelet, Riesz terem, Aradi Vértanúk tere 1., Szeged

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