In this paper, we first introduce the pseudo-entropy of free semigroup actions and show that it is equal to topological entropy defined by Bufetov [9]. Second, for actions, concepts chain recurrence time, mixing time are introduced, upper bounds these times calculated. Furthermore, lower box dimension provide a bound on actions. Third, structure transitive systems discussed. Our analysis genera...

Z. Feng and R.W. Heath proved that any separable linearly ordered space (LOTS) which is a cancellative topological semigroup must be metrizable. In this note, we show that the same holds more generally for CCC LOTS by proving that no Souslin line admits a continuous cancellative binary operation. We also show that no Lindelöf Aronszajn line admits such an operation.

We develop some new topological tools to study maximal subgroups of free idempotent generated semigroups. As an application, we show that the rank 1 component of the free idempotent generated semigroup of the biordered set of a full matrix monoid of size n×n,n > 2 over a division ring Q has maximal subgroup isomorphic to the multiplicative subgroup of Q.

We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem. Furstenberg and his co-workers have shown [15, 16, 17] how to deduce combinatorial consequences from theorems about topological dynamics in compact metric spaces. Bergelson and ...

Margolis, S.W. and J.E. Pin, New results on the conjecture of Rhodes and on the topological conjecture, Journal of Pure and Applied Algebra 80 (1992) 305-313. The Conjecture of Rhodes, originally called the ‘type II conjecture’ by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems...

A topological machine is a topo |ogized quas i -machine of G i n s b u r g and is defined by two cont inuous funct ions f and g, T ~ X x S ~ X , the s ta tet rans i t ion funct ion and the output funct ion, respect ively, where .X is a nonvoid T2-space, the state space, S and T are two topological semigroups , the i n p u t semigroup and the ou tpu t semigroup, respect ively, such tha t f (x, s...

A left order on a magma (e.g., semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles...