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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
Solid weak BCC-algebras
We characterize weak BCC-algebras in which the identity is
satisfied only in the case when elements belong to the same branch
Presentations of Wess-Zumino-Witten Fusion Rings
The fusion rings of Wess-Zumino-Witten models are re-examined. Attention is
drawn to the difference between fusion rings over Z (which are often of greater
importance in applications) and fusion algebras over C. Complete proofs are
given characterising the fusion algebras (over C) of the SU(r+1) and Sp(2r)
models in terms of the fusion potentials, and it is shown that the analagous
potentials cannot describe the fusion algebras of the other models. This
explains why no other representation-theoretic fusion potentials have been
found.
Instead, explicit generators are then constructed for general WZW fusion
rings (over Z). The Jacobi-Trudy identity and its Sp(2r) analogue are used to
derive the known fusion potentials. This formalism is then extended to the WZW
models over the spin groups of odd rank, and explicit presentations of the
corresponding fusion rings are given. The analogues of the Jacobi-Trudy
identity for the spinor representations (for all ranks) are derived for this
purpose, and may be of independent interest.Comment: 32 pages, 3 figures, added references, minor additions to text. To be
published in Rev. Math. Phy
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