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A new scheme for proving pseudoidentities from a given set {Sigma} of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when {Sigma} defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when {Sigma} defines a pseudovariety V which is {sigma}-reducible for the equation x=y, provided {Sigma} is enough to prove a basis of identities for the variety of {sigma}-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.
Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of Godel logic. The calculi are used to establish completeness and complexity results for these fragments.
Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated groups $A$.
Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $log |G| / log n leq b(G) < 45 (log |G| / l
We give a new proof of Gromovs theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was