176 research outputs found
Atom structures of cylindric algebras and relation algebras
Accepted versio
Axiomatizability of reducts of algebras of relations
Submitted versio
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
Cylindric skew Schur functions
Cylindric skew Schur functions, which are a generalisation of skew Schur
functions, arise naturally in the study of P-partitions. Also, recent work of
A. Postnikov shows they have a strong connection with a problem of considerable
current interest: that of finding a combinatorial proof of the non-negativity
of the 3-point Gromov-Witten invariants. After explaining these motivations, we
study cylindric skew Schur functions from the point of view of
Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we
generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus
giving an expansion of an arbitrary cylindric skew Schur function in terms of
skew Schur functions. While we show that no non-trivial cylindric skew Schur
functions are Schur-positive, we conjecture that this can be reconciled using
the new concept of cylindric Schur-positivity.Comment: 32 pages, 14 figures. Minor expository improvements. Version to
appear in Advances in Mathematic
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
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