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1.
A hierarchy of partial abelian structures is considered. In an order of decreasing generality, these structures include partial
abelian monoids (PAM), cancellative PAMs (CPAM), effect algebras (or D-posets), orthoalgebras, orthomodular posets (OMP) and
orthomodular lattices (OML). If P is a PAM, the concepts of a congruence on P and a quotient P
are defined. Similar definitions are given for quotients of higher level categories in the hierarchy. The notion of a Riesz
ideal I on a CPAM P is defined and it is shown that I generates a congruence on P. The corresponding quotients P/I for categories in the hierarchy are studied. It is shown that a subset I of an OML is a Riesz ideal if and only if I is a p-ideal. Moreover, for effect algebras, we show that congruences generated by Riesz ideals are precisely those that
are given by a perspectivity. The paper includes a large number of counterexamples and examples that illustrate various concepts.
Received April 14, 1997; accepted in final form January 19, 1998. 相似文献
2.
Bernhard Schmidt 《Journal of Algebraic Combinatorics》1997,6(3):279-297
This paper provides new exponent and rank conditions for the existence of abelian relative (p
a,p
b,p
a,p
a–b)-difference sets. It is also shown that no splitting relative (22c,2d,22c,22c–d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G Z8 × Z4 × Z2 exists if and only if exp(G) 4 or G = Z8 × (Z2)3 with N Z2 × Z2. 相似文献
3.
In this article, finite p-groups all of whose proper quotient groups are abelian or inner-abelian are classified. As a corollary, finite p-group all of whose proper quotient groups are abelian, and finite p-groups all of whose proper sections are abelian or inner-abelian are also classified. 相似文献
4.
Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (22d+2, 22d+1±2
d
, 22d
±2
d
). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2
d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 22d+2 with exponent 2
d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case. 相似文献
5.
6.
Helge Møller Pedersen 《Geometriae Dedicata》2018,195(1):283-305
Given a rational homology 3-sphere M whose splice diagram \(\varGamma (M)\) satisfies the semigroup condition, Neumann and Wahl define a complete intersection surface singularity called a splice diagram singularity. Under an additional hypothesis on M called the congruence condition they show that the link of this singularity is the universal abelian cover of M. They ask if this still holds if the congruence condition fails. In this article we generalize the congruence condition to orientable graph orbifolds. We show that under a small additional hypothesis this orbifold congruence condition implies that the link of the splice diagram singularity is the universal abelian cover. By showing that any two-node splice diagram satisfying the semigroup condition is the splice diagram of an orbifold satisfying the orbifold congruence condition, we answer the question of Neumann and Wahl affirmatively for two-node diagrams. However, examples show this approach to their question no longer works for three nodes. 相似文献
7.
There is a natural way to associate to any tree T with leaf set X, and with edges weighted by elements from an abelian group G, a map from the power set of X into G—simply add the elements on the edges that connect the leaves in that subset. This map has been well-studied in the case where
G has no elements of order 2 (particularly when G is the additive group of real numbers) and, for this setting, subsets of leaves of size two play a crucial role. However, the existence and uniqueness results in that setting do not extend to arbitrary abelian groups.
We study this more general problem here, and by working instead with both, pairs and triples of leaves, we obtain analogous existence and uniqueness results. Some particular results for elementary abelian 2-groups
are also described.
Received July 13, 2005 相似文献
8.
Darryn Bryant Judith Egan Barbara Maenhaut Ian M. Wanless 《Designs, Codes and Cryptography》2009,50(1):93-105
In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special
class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial
difference set respectively in two groups of orders v
1 and v
2 with |v
1 − v
2| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in
the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference
set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary
abelian group of order 35 or 37, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 39. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other
when q ≤ 311.
相似文献
9.
Angelina Y. M. Chin 《Proceedings Mathematical Sciences》2009,119(2):145-148
Let R be a ring with identity. An element in R is said to be clean if it is the sum of a unit and an idempotent. R is said to be clean if all of its elements are clean. If every idempotent in R is central, then R is said to be abelian. In this paper we obtain some conditions equivalent to being clean in an abelian ring. 相似文献
10.
We call a group G with subgroups G1, G2 such that G = G1G2 and both N = G1 ∩ G2 and G1 are normal in G a semidirect product with amalgamated subgroup N. We show that if Gl is a group with Nl ? Gl containing a relative ‐difference set relative to Nl for l = 1,2, and if there exists a “compatible coupling” from (G2, N2) to (G1, N1), a notion introduced in the paper, then for any i,j ∈ ? there exists at least one semidirect product with amalgamated subgroup N ? N1 ? N2 containing a relative ‐difference set. We say “at least one” to emphasize that the proof is via recursive construction and that different groups may be obtained depending on the choices made at different stages of the recursion. A special case of this result shows that if K is any finite group containing a normal relative ‐difference set, then there exists, for each i ∈ ?, at least one semidirect product with amalgamated subgroup N containing a relative ‐difference set. These results suggest that the class of semidirect products with an amalgamated subgroup provides a rich source of new (non‐abelian) semiregular relative difference sets. © 2004 Wiley Periodicals, Inc. 相似文献