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Tensor products of distributive lattices and their Priestley duals

Tensor products of distributive lattices and their Priestley duals Acta Mathematica Academiae Scientiarum Hungaricae Tomus 35 (3--4), (1980), 387--392. TENSOR PRODUCTS OF DISTRIBUTIVE LATTICES AND THEIR PRIESTLEY DUALS By J. SCHMID (Bern) O. Introduction Tensor products of distributive lattices and of semilattices have been considered by several authors, namely in a series of papers by FI~AS~R [3], [41, [5] and [6], but also in [7], [8] and [9]. Tensor products -- defined as universal objects through which every bihomomorphism splits -- exist in any variety of algebras. This note deals with tensor products in the variety D of distributive lattices and in the variety S of distributive join-semilattices. The former is called lattice tensor product and written LI| the latter semilattice tensor product, written SI| Existence and uniqueness are established in the usual way, taking suitable homomorphic images of the corresponding free structures generated by the cartesian product of the factors. See [3] and [4] for details. It is somewhat surprising that for L1, LzCD their semilattice tensor product LI| 2 (obtained by considering the L~ as join-semilattices) is in fact even a (distributive) lattice (Theorem 2.6 in [3]). So there are two kinds of tensor product available within D" The main purpose of this note is to http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Tensor products of distributive lattices and their Priestley duals

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Springer Journals
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Copyright
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Mathematics; Mathematics, general
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0001-5954
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1588-2632
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10.1007/BF01886308
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Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 35 (3--4), (1980), 387--392. TENSOR PRODUCTS OF DISTRIBUTIVE LATTICES AND THEIR PRIESTLEY DUALS By J. SCHMID (Bern) O. Introduction Tensor products of distributive lattices and of semilattices have been considered by several authors, namely in a series of papers by FI~AS~R [3], [41, [5] and [6], but also in [7], [8] and [9]. Tensor products -- defined as universal objects through which every bihomomorphism splits -- exist in any variety of algebras. This note deals with tensor products in the variety D of distributive lattices and in the variety S of distributive join-semilattices. The former is called lattice tensor product and written LI| the latter semilattice tensor product, written SI| Existence and uniqueness are established in the usual way, taking suitable homomorphic images of the corresponding free structures generated by the cartesian product of the factors. See [3] and [4] for details. It is somewhat surprising that for L1, LzCD their semilattice tensor product LI| 2 (obtained by considering the L~ as join-semilattices) is in fact even a (distributive) lattice (Theorem 2.6 in [3]). So there are two kinds of tensor product available within D" The main purpose of this note is to

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Jun 15, 2005

References

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    Balbes, R.
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    Balbes, R.; Dwinger, Ph.
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    Fraser, G.
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    Fraser, G.
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    Fraser, G.
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    Fraser, G.
  • Congruence lattices of tensor products of lattices. Abstract 78T-A149
    Grätzer, G.; Quackenbush, R. W.
  • The semilattice tensor product of projective distributive lattices. Abstract 78T-A-148
    Lakser, H.
  • Representation of distributive lattices by means of ordered STone spaces
    Priestley, H. A.
  • Ordered topological spaces and the representation of distributive lattices
    Priestley, H. A.