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A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xyin{x,y}$ ($xy otin{x,y}$) for any (distinct) elements $x,yin S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $sqrt[infty]{e}={xin S:exists ninmathbb N;;(x^n=e)}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected Cayley graph
We compute the cogrowth series for Baumslag-Solitar groups $mathrm{BS}(N,N) = < a,b | a^N b = b a^N > $, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all
The field of in-vivo neurophysiology currently uses statistical standards that are based on tradition rather than formal analysis. Typically, data from two (or few) animals are pooled for one statistical test, or a significant test in a first animal
Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the b