Birkhoff Completeness in Institutions

Three decades of institution theory

institution to a system of institutions, i.e. a diagram of institutions, more precisely. However the thinking along these lines has led to a rather different and much more efficient solution, namely that of the ‘flattening’ of the respective diagram of institutions to a single institution by extending a corresponding construction from category theory [49] to institutions. The resulting concept ...

On the Completeness Theorem of Many-sorted Equational Logic and the Equivalence between Hall Algebras and Bénabou Theories

The completeness theorem of equational logic of Birkhoff asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for many-sorted deduction, founded ultimately on the congruences on Hall algebras, generalized the completeness theorem of Birkhoff to the completeness theorem of many-sorted ...

Spectrality of Ordinary Differential Operators

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford’s spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein’s conjecture that dissipative differential operators are Birkhoff-regular and give sharp estimate of the norms of spectral projectors in the odd case. Considerations are based on ...

The Completeness Theorem for Monads in Categories of Sorted Sets

Birkhoff’s completeness theorem of equational logic asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for finitary many-sorted equational deduction, generalized the completeness theorem of Birkhoff to the completeness of finitary many-sorted equational logic and provided simultaneo...

Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem