In this paper, a commutative semigroup will be written as a disjoint union of its cancellative subsemigroups. Based on this fact we will define the left regular representation of a commutative separative semigroup and show that this representation is faithful. Finally concrete examples of commutative separative semigroups, their decompositions and their left regular representations are given.

We generalize a theorem of Frey by giving sufficient conditions for a subsemigroup T of a cancellative left amenable semigroup S to be left amenable. In particular, we show that if S is left amenable and T does not contain a free subsemigroup on two generators, then T is left amenable as well.

We show that David Rees’ 1948 paper on the structure of left cancellative monoids stands at the threshold of the theory of self-similar group actions, and in doing so correct an error in a paper by Nivat and Perrot. 2000 AMS Subject Classification: 20M05, 20M12, 20M18.

In this note we will show that the dilation result obtained for fractional skew monoid rings, in the case of a cancellative left Ore monoid S acting on a unital ring A by corner isomorphisms, holds in full generality. We apply this result to the context of semigroup C∗-crossed products.

Suppose A, S, and T are semigroups, e: A → S and f : A → T semigroup homomorphisms, and X a generating set for S. We assume (1) that every element of S divides some element of e(A), (2) that T is cancellative, (3) that T is power-cancellative (i.e, xd = yd ⇒ x = y for d > 0), and (4) a further technical condition, which in particular holds if T admits a semigroup ordering with the order-type of...

We develop alternative semantics for à Lukasiewicz logic and for cancellative hoop logic according to the following idea. We formalize statements reflecting an inexact knowledge of certain (sharp) properties; we assume that all what can be known about a property is its expressive strength. To this end, we consider a Boolean algebra endowed with an automorphism group or, alternatively, with a me...

Recently the question of standard completeness of MTL and its extensions were deeply studied. Jenei and Montagna showed that MTL logic satisfies standard completeness theorem (see [6]). In [2], the authors studied standard completeness of ΠMTL which is an axiomatic extension of MTL logic where the conjunction is interpreted by a left-continuous cancellative t-norm. They showed that ΠMTL logic i...

Notice that the upper bound is an immediate consequence of (I<;J) :s;; 2n. To prove the lower bound we use an algebraic construction which is a modification of a construction of Babai and S6s [1]. How a family of sets can fail to have the union-free property? There are essentially two possibilities: (a) there are four distinct sets A, B, C, D with Au B =CuD. (b) there are three distinct sets A,...