From 269f872019b616287c4cf1878477879019f8bbd0 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 15 Nov 2019 12:44:03 -0500 Subject: [PATCH] eja: rename is_minimal_idempotent() to is_primitive_idempotent(). The term "primitive" is more common in optimization, anyway. Also the word "minimal" is a bit overloaded. Oh, and I added two more tests to preclude zero from being primitive. --- mjo/eja/eja_element.py | 42 +++++++++++++++++++++++++++++++++++------- 1 file changed, 35 insertions(+), 7 deletions(-) diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 4d3b071..276eab0 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -546,10 +546,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): return not (p(zero) == zero) - def is_minimal_idempotent(self): + def is_primitive_idempotent(self): """ - Return whether or not this element is a minimal idempotent. + Return whether or not this element is a primitive (or minimal) + idempotent. + A primitive idempotent is a non-zero idempotent that is not + the sum of two other non-zero idempotents. Remark 2.7.15 in + Baes shows that this is what he refers to as a "minimal + idempotent." An element of a Euclidean Jordan algebra is a minimal idempotent if it :meth:`is_idempotent` and if its Peirce subalgebra @@ -560,6 +565,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: RealSymmetricEJA, + ....: TrivialEJA, ....: random_eja) WARNING:: @@ -575,7 +581,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x = sum(J.gens()) sage: x.is_regular() False - sage: [ c.is_minimal_idempotent() + sage: [ c.is_primitive_idempotent() ....: for (l,c) in x.spectral_decomposition() ] [False, True] @@ -586,7 +592,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) ) sage: x.is_regular() True - sage: [ c.is_minimal_idempotent() + sage: [ c.is_primitive_idempotent() ....: for (l,c) in x.spectral_decomposition() ] [True, True] @@ -596,7 +602,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = random_eja() - sage: J.rank() == 1 or not J.one().is_minimal_idempotent() + sage: J.rank() == 1 or not J.one().is_primitive_idempotent() True A non-idempotent cannot be a minimal idempotent:: @@ -604,7 +610,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = JordanSpinEJA(4) sage: x = J.random_element() - sage: (not x.is_idempotent()) and x.is_minimal_idempotent() + sage: (not x.is_idempotent()) and x.is_primitive_idempotent() False Proposition 2.7.19 in Baes says that an element is a minimal @@ -615,14 +621,36 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: J = JordanSpinEJA(4) sage: x = J.random_element() sage: expected = (x.is_idempotent() and x.trace() == 1) - sage: actual = x.is_minimal_idempotent() + sage: actual = x.is_primitive_idempotent() sage: actual == expected True + Primitive idempotents must be non-zero:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.zero().is_idempotent() + True + sage: J.zero().is_primitive_idempotent() + False + + As a consequence of the fact that primitive idempotents must + be non-zero, there are no primitive idempotents in a trivial + Euclidean Jordan algebra:: + + sage: J = TrivialEJA() + sage: J.one().is_idempotent() + True + sage: J.one().is_primitive_idempotent() + False + """ if not self.is_idempotent(): return False + if self.is_zero(): + return False + (_,_,J1) = self.parent().peirce_decomposition(self) return (J1.dimension() == 1) -- 2.44.2