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Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations
A partial algebra construction of Gr\"atzer and Schmidt from
"Characterizations of congruence lattices of abstract algebras" (Acta Sci.
Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to
a well-known fact that every finite distributive lattice is representable, seen
as a special case of the Finite Lattice Representation Problem.
The construction of this proof brings together Birkhoff's representation
theorem for finite distributive lattices, an emphasis on boolean lattices when
representing finite lattices, and a perspective based on inequalities of
partially ordered sets. It may be possible to generalize the techniques used in
this approach.
Other than the aforementioned representation theorem only elementary tools
are used for the two theorems of this note. In particular there is no reliance
on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and
Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice
to be representable (see William J. DeMeo), such as the closure method
A short note on a Bernstein-Bezier basis for the pyramid
We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to
the face reduces to the Bernstein-Bezier basis on the triangle or
quadrilateral. The basis satisfies the standard positivity and partition of
unity properties common to Bernstein polynomials, and spans the same space as
non-polynomial pyramid bases in the literature.Comment: Submitte
New Angle on the Strong CP and Chiral Symmetry Problems from a Rotating Mass Matrix
It is shown that when the mass matrix changes in orientation (rotates) in
generation space for changing energy scale, then the masses of the lower
generations are not given just by its eigenvalues. In particular, these masses
need not be zero even when the eigenvalues are zero. In that case, the strong
CP problem can be avoided by removing the unwanted term by a chiral
transformation in no contradiction with the nonvanishing quark masses
experimentally observed. Similarly, a rotating mass matrix may shed new light
on the problem of chiral symmetry breaking. That the fermion mass matrix may so
rotate with scale has been suggested before as a possible explanation for
up-down fermion mixing and fermion mass hierarchy, giving results in good
agreement with experiment.Comment: 14 page
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