The concept of the direct product decomposition (DPD) is extended to arbitrary tensors while maintaining the same theoretical reduction in storage and computation. Additionally, the structure of the ...DPD as introduced by Gauss and Stanton is shown to be but one of a family of direct product decompositions which may be visualised using graphs. One particular member of this family is also shown to be critically important in relating the DPD and symmetry blocking approaches. Lastly, an implementation of tensor contraction using this extended DPD based on recent work in dense tensor contraction is presented, showing how the particular DPD used to represent the tensors in memory or on disk may be divorced from the optimal DPD used for a particular tensor contraction. The performance of the new algorithm is benchmarked by interfacing with the CFOUR programme suite, where significant speedups for CCSD calculations are observed.
Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be ...useful in the study of direct products.
Abstract
We introduce a non-commutative generalization of quasi-MV algebra, called quasipseudo-MV algebra. We present some properties of quasi-pseudo-MV algebras and investigate the direct product ...decomposition of them. Further, we generalize quasi
l
-group to the non-commutative case and prove the interval of a non-commutative quasi
l
-group with a strong quasi-unit is a quasipseudo-MV algebra.
Locally adequate semigroup algebras Ji, Yingdan; Luo, Yanfeng
Open mathematics (Warsaw, Poland),
1/2016, Volume:
14, Issue:
1
Journal Article
Peer reviewed
Open access
We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup ...algebra into a direct product of primitive abundant
-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the
-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.
In Harding (Trans. Amer. Math. Soc.
348
(5), 1839–1862
1996
) it was shown that the direct product decompositions of any non-empty set, group, vector space, and topological space
X
form an ...orthomodular poset Fact
X
. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. Here we develop dynamics and an abstract version of a time independent Schrödinger’s equation in the setting of decompositions by considering representations of the group of real numbers in the automorphism group of the orthomodular poset Fact
X
of decompositions.
It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by ...the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics.
The closed subspaces
of a Hilbert space
correspond to direct product decompositions
of the Hilbert space, a result that lies at the heart of the superposition principle. In
it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set
, the automorphism group of the orthomodular poset Fact(
) of direct product decompositions of
is isomorphic to the permutation group of
The structure Fact(
) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(
) are determined in proving our version of Wigner’s theorem, namely that Fact(
) is atomistic in a very strong way.
Automorphisms of decompositions Hannan, Tim; Harding, John
Mathematica Slovaca,
04/2016, Volume:
66, Issue:
2
Journal Article
Peer reviewed
Open access
In HARDING, J.:
, Trans. Amer. Math. Soc.
(1996), 1839–1862 the author showed that the direct product decompositions of many different types of structures, such as sets, groups, vector spaces, ...topological spaces, and relational structures, naturally form orthomodular posets. When applied to the direct product decompositions of a Hilbert space, this construction yields the familiar orthomodular lattice of closed subspaces of the Hilbert space.
In this note we consider orthomodular posets Fact
of decompositions of a finite set
. We consider the structure of these orthomodular posets, such as their size, shape, connectedness, states, and begin a study of their automorphism groups in the context of the natural map Γ from the group of permutations of
to the automorphism group of F
We show Γ is an embedding except when |
| is prime or 4, and completely describe the situation when |
| has two or fewer prime factors, when |
| = 2
and when |
| = 3
. The bulk of our effort lies in a series of combinatorial arguments to show Γ is an isomorphism when |
| = 27. We conjecture that this is the case whenever |
| has sufficiently many prime factors of sufficient size, and hope that our arguments here might be adapted to the general case.
We apply the concept of generalized MV-algebra (GMV-algebra, for short) in the sense defined and studied by Galatos and Tsinakis. We introduce the notion of isometry of a GMV-algebra; we investigate ...the relations between isometries and direct product decompositions of GMV-algebras. Using these relations we show that if a GMV-algebra
M
has a greatest element, then each isometry of
M
is idempotent.
In this paper, we further study the pseudo-t-norms and implication operators on a complete Brouwerian lattice, and discuss their direct products and direct product decompositions.
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