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1.
C. J. van Alten 《Algebra Universalis》2007,57(1):47-62
The implicational subreducts of n-potent commutative integral residuated lattices are axiomatized using a new embedding of a BCK-algebra into a commutative
integral residuated lattice. The class of {→, 1, ≤ }-subreducts of commutative residuated lattices satisfying x
n
≤ x
m
is also axiomatized, as are other subreduct classes.
Presented by R. W. Quackenbush.
Received February 3, 2006; accepted in final form May 12, 2006. 相似文献
2.
A. V. Kartashova 《Journal of Mathematical Sciences》2009,163(6):682-687
In this paper, it is shown that the dual
[(\textQord)\tilde]\mathfrakA \widetilde{\text{Qord}}\mathfrak{A} of the quasiorder lattice of any algebra
\mathfrakA \mathfrak{A} is isomorphic to a sublattice of the topology lattice
á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . Further, if
\mathfrakA \mathfrak{A} is a finite algebra, then
[(\textQord)\tilde]\mathfrakA @ á( \mathfrakA ) \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) . We give a sufficient condition for the lattices
[(\textCon)\tilde]\mathfrakA\text, [(\textQord)\tilde]\mathfrakA \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} , and
á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras. 相似文献
3.
Miroslav Ploščica 《Algebra Universalis》2003,49(1):1-12
4.
We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with
and
and let A = ‖a
ij
‖
n×n
, where a
ij
∈ P for i, j = 1,..., n. Let A* = ‖a
ij
′
‖
n×n
and
for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤).
Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL
n
(P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) −
, ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL
a
(P, ≤) ≅ = S
n
k
.
We give some further results concerning inversion of matrices over a pseudocomplemented lattice.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005. 相似文献
5.
The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices. 相似文献
6.
7.
Let ${{\mathcal D}}$ be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d opp} is definable, where d and d opp are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identity automorphism of ${{\mathcal D}}$ is the opposite map. Then we apply these results to investigate definability in the closely related lattice of universal classes of distributive lattices. We prove that this lattice has only one non-identity automorphism, the opposite map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each element K of the two subsets, {K, K opp} is a definable subset of the lattice. 相似文献
8.
In this paper we give a necessary and sufficient condition for a groupoid D such that the sup-min product is distributive over arbitrary intersection of fuzzy subsets of D, and correct some results from the paper [S. Ray, The lattice of all idempotent fuzzy subsets of a groupoid, Fuzzy Sets and Systems 96 (1998) 239–245]. Also, we prove that the set of all idempotent fuzzy sets forms a complete lattice, which is a complete join-sublattice of the lattice of all fuzzy subgroupoids. This result extends the corresponding result from the above mentioned paper. 相似文献
9.
Let be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and is represented by the restriction map from Con K to Con L. The particular case where is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter. 相似文献
10.
11.
For a monounary algebra (A, f) we denote R
∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R
∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R
∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary
algebras. 相似文献
12.
We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually
characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras
of bounded distributive lattices by means of Priestley quasi-orders (Adams, Algebra Univers 3:216–228, 1973; Cignoli et al., Order 8(3):299–315, 1991; Schmid, Order 19(1):11–34, 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually
characterized by Vietoris families. We show that this generalizes the well-known characterization (Priestley, Proc Lond Math
Soc 24(3):507–530, 1972) of homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show
how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of
subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations
of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147–151, 1974). 相似文献
13.
Harry Lakser 《Algebra Universalis》1981,13(1):78-81
Grant A. Fraser defined the semilattice tensor productA ⊗B of distributive latticesA, B and showed that it is a distributive lattice. He proved that ifA ⊗B is projective then so areA andB, that ifA andB are finite and projective thenA ⊗B is projective, and he gave two infinite projective distributive lattices whose semilattice tensor product is not projective.
We extend these results by proving that ifA andB are distributive lattices with more than one element thenA ⊗B is projective if and only if bothA andB are projective and both have a greatest element.
Presented by W. Taylor. 相似文献
14.
The set of all m × n Boolean matrices is denoted by $
\mathbb{M}
$
\mathbb{M}
m,n
. We call a matrix A ∈ $
\mathbb{M}
$
\mathbb{M}
m,n
regular if there is a matrix G ∈ $
\mathbb{M}
$
\mathbb{M}
n,m
such that AGA = A. In this paper, we study the problem of characterizing linear operators on $
\mathbb{M}
$
\mathbb{M}
m,n
that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $
\mathbb{M}
$
\mathbb{M}
m,n
strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $
\mathbb{M}
$
\mathbb{M}
m,n
strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $
\mathbb{M}
$
\mathbb{M}
m,n
, or m = n and T(X) = UX
T
V for all X ∈ $
\mathbb{M}
$
\mathbb{M}
n
. 相似文献
15.
Let ℝ
n
be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed
sphere in ℝ
n
of radius $
\sqrt {n/4}
$
\sqrt {n/4}
contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. 相似文献
16.
Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity
for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesI
n, An, Dn, En and their duals for small values ofn. Although it appears thatCM(I
n)=2
n−1 ifn≤33, asn → ∞ we haveCM(I
n)∼2.089...
n
. The results have application to numerical integration. 相似文献
17.
We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~... 相似文献
18.
Summary G. Grätzer and H. Lakser proved in 1986 that for the finite distributive lattices D and E, with |D| > 1, and for the {0, 1}-homomorphism φ of D into E, there exists a finite lattice L and an ideal I of L such that D ≡ Con L, E ≡ Con I, and φ is represented by the restriction map. In their recent survey of finite congruence lattices, G. Grätzer and E. T. Schmidt ask whether this result can be improved by requiring that L be sectionally complemented. In this note, we provide an affirmative answer. The key to the solution is to generalize the 1960 sectional complement (see Part I) from finite orders to finite preorders. 相似文献
19.
The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely,
`A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for
any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements
occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with
n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
George M. Bergman 《Algebra Universalis》2005,53(2-3):357-395
Properties of several sorts of lattices of convex subsets of
are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of
and the lattice of all convex subsets of
The lattices of arbitrary, of open bounded, and of compact convex sets in
all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set
satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of
To the memory of Ivan RivalReceived April 22, 2003; accepted in final form February 16, 2005.This revised version was published online in August 2005 with a corrected cover date. 相似文献