One dimensional topological lattices

Anderson, Lee

doi:10.1090/s0002-9939-1959-0107678-7

Cited by 20 publications

(11 citation statements)

References 8 publications

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“…Thus L is locally compact. It is known [5] that a locally compact connected topological lattice is locally convex and so (iv) implies (i) which completes the proof.…”

Section: Lemma 1 If L Is a Topological Lattice And If A Is A Subset supporting

confidence: 57%

“…In particular, Lemma 2 implies that a locally convex connected topological lattice is locally connected. It has already been shown [5] that a locally compact connected topological lattice is locally convex so that we have the Theorem 2. A locally compact connected topological lattice is locally connected.…”

Section: Lemma 1 If L Is a Topological Lattice And If A Is A Subset mentioning

confidence: 71%

“…We will first prove that mA(a\/L) is a chain. Since a^b, it follows that m=a\/b is in F(a\/L) (see [5]). As in the proof of Lemma 2 we have that cd(F(aVL))^l and so cd(mA(oVI)) ^1.…”

Section: If We Define 77: 7x7->l By H(t T') = F(t) Ah(t') Then Clearmentioning

confidence: 98%

“…Now 73 is a closed subset of L and therefore is locally compact. Moreover, 73 is a connected sublattice of L and so 73 is a chain (see [5]). Now x and y are elements of 73 and so xAy =x or y which implies that a\/L is a chain.…”

Section: Lemma 1 If L Is a Topological Lattice And If A Is A Subset mentioning

confidence: 99%

“…Remark. It is known [5] that a locally compact connected topological lattice is a chain if, and only if, it is at most one-dimensional.…”

Section: If We Define 77: 7x7->l By H(t T') = F(t) Ah(t') Then Clearmentioning

confidence: 99%

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On the distributivity and simple connectivity of plane topological lattices

Anderson¹

1959

Trans. Amer. Math. Soc.

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“…Thus L is locally compact. It is known [5] that a locally compact connected topological lattice is locally convex and so (iv) implies (i) which completes the proof.…”

Section: Lemma 1 If L Is a Topological Lattice And If A Is A Subset supporting

confidence: 57%

Section: Lemma 1 If L Is a Topological Lattice And If A Is A Subset mentioning

confidence: 71%

“…We will first prove that mA(a\/L) is a chain. Since a^b, it follows that m=a\/b is in F(a\/L) (see [5]). As in the proof of Lemma 2 we have that cd(F(aVL))^l and so cd(mA(oVI)) ^1.…”

Section: If We Define 77: 7x7->l By H(t T') = F(t) Ah(t') Then Clearmentioning

confidence: 98%

Section: Lemma 1 If L Is a Topological Lattice And If A Is A Subset mentioning

confidence: 99%

“…Remark. It is known [5] that a locally compact connected topological lattice is a chain if, and only if, it is at most one-dimensional.…”

Section: If We Define 77: 7x7->l By H(t T') = F(t) Ah(t') Then Clearmentioning

confidence: 99%

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On the distributivity and simple connectivity of plane topological lattices

Anderson¹

1959

Trans. Amer. Math. Soc.

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Homomorphisms and dimension

Anderson¹,

Hunter²

1962

Math. Ann.

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The present work is devoted for the most part to the following general question: Let S be a compact connected n-dimensional semigroup with identity, and let / be a continuous homomorphism onto the 8emigroup T. What can be said about the dimension o/T ? It is well known that i/S is a locally compact group thenAnd, as we shall see, if S is a compact connected lattice and T is finite dimensional dim T < dims.However, there exists a compact connected one dimensional semigroup with identity S and a continuous homomorphism / such that /(S) is infinite dimensional. That is to say, a continuous homomorphism may be dimension raising. In connection with this however, the following is known: Let S be a compact, connected, locally connected one dimensional semigroup with identity. Then S admits no dimension raising homomorphism. (See [13].)The paper is in four sections. In the first we establish a number of preliminary lemmas for later use. We also establish the existence of various kinds of dimension raising homomorphisms. In particular, we establish the existence of monotone open, dimension raising homomorphisms.In the second section, we consider certain dimensionally stable semigroups. That is to say, semigroups whose dimension cannot be raised by a continuous homomorphism. Specifically, we establish the dimensional stability of certain algebraically irreducible semigroups. Indeed, we show that every compact connected abelian semigroup with identity contains a compact connected subsemigroup containing the minimal ideal and the identity which admits no dimension raising continuous homomorphism. Thus, in such a situation, under a homomorphism, there is always available a particular sub-semigroup whose dimension is not raised.In the third section we consider the effect upon dimension of monotone homomorphisms. We show that the dimension of a one dimensional compact connected semigroup with identity cannot be raised by either a monotone or an open homomorphism. We also consider the related problem of the immersability of the image of a compact connected plane semigroup under a monotone homomorphism.

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Stone MV-algebras and strongly complete MV-algebras

Nganou

2017

Algebra Univers.

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Abstract. The article has two main objectives: characterize compact Hausdorff topological MV-algebras and Stone MV-algebras on one hand, and characterize strongly complete MV-algebras on the other hand. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies [0, 1] with the interval topology and finite Lukasiewicz chains with discrete topology. Going one step further we also prove that Stone MV-algebras are product (both topological and algebraic) of finite Lukasiewicz chains with discrete topology. In the second part we prove that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and its only maximal ideals of finite ranks are principal.To the memory of a great mentor, John V. Leahy (1937Leahy ( -2015.

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One dimensional topological lattices

Cited by 20 publications

References 8 publications

On the distributivity and simple connectivity of plane topological lattices

On the distributivity and simple connectivity of plane topological lattices

Homomorphisms and dimension

Stone MV-algebras and strongly complete MV-algebras

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