We describe a correspondence between a class of left cancellative monoids and self-similar group actions in the sense of Nekrashevych et al. This correspondence originated in Perrot’s 1972 thesis, and developed the ideas to be found in Rees’ 1948 paper. 2000 AMS Subject Classification: 20M10, 20M50.

We construct reduced and full semigroup C*-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz. Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C*-algebras in analogy to the group case.

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.

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,...