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  • 1
    Electronic Resource
    Electronic Resource
    Intrinsic Spectral Topologies (1994)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 728 (1994), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Notes: : Given a poset (X; ≤), any spectral topology with specialisation order ≤ lies between the upper topology and the Scott topology. Conditions are given under which these extreme topologies are indeed spectral. The results contribute to the understanding of speectral sets, that is, posets which arise as the prime ideals of commutative rings with 1, ordered by inclusion, or equivalently as the prime ideals of distributive lattices with 0 and 1, ordered by inclusion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1994.tb44135.x
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  • 2
    Electronic Resource
    Electronic Resource
    Optimal natural dualities for varieties of Heyting algebras (1996)
    Springer
    Studia logica 56 (1996), S. 67-96 
    Details
    ISSN: 1572-8730
    Keywords: Heyting algebra ; natural duality ; optimal duality ; 06D20 ; 06D05 ; 08B99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00370141
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  • 3
    Electronic Resource
    Electronic Resource
    On Priestley Spaces of Lattice-Ordered Algebraic Structures (1998)
    Springer
    Order 15 (1998), S. 297-323 
    Details
    ISSN: 1572-9273
    Keywords: duality ; implicative lattice ; lattice-ordered group ; Priestley space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The laws defining many important varieties of lattice-ordered algebras, such as linear Heyting algebras, MV-algebras and l-groups, can be cast in a form which allows dual representations to be derived in a very direct, and semi-automatic, way. This is achieved by developing a new duality theory for implicative lattices, which encompass all the varieries above. The approach focuses on distinguished subsets of the prime lattice filters of an implicative lattice, ordered as usual by inclusion. A decomposition theorem is proved, and the extent to which the order on the prime lattice filters determines the implicative structure is thereby revealed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1006224930256
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  • 4
    Electronic Resource
    Electronic Resource
    Series-parallel posets and relative Ockham lattices (1994)
    Springer
    Order 11 (1994), S. 281-305 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; 06D05 ; Series-parallel poset ; order-reversing map ; distributive lattice ; Ockham algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A study is undertaken of order-reversing maps on series-parallel posets and structural characterisations are obtained of various subclasses of such ordered sets. The results are applied to complete the authors' earlier investigation of classes $$\mathcal{A}^{rel} $$ of finite relate $$\mathcal{A}$$ lattices, where $$\mathcal{A}$$ is a variety of Ockham lattices and the distributive lattice duals of the algebras in $$\mathcal{A}^{rel} $$ are required to be series-parallel.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02115816
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  • 5
    Electronic Resource
    Electronic Resource
    Books in review (1984)
    Springer
    Order 1 (1984), S. 205-208 
    Details
    ISSN: 1572-9273
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00565654
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  • 6
    Electronic Resource
    Electronic Resource
    Natural dualities for varieties ofn-valued łukasiewicz algebras (1995)
    Springer
    Studia logica 54 (1995), S. 333-370 
    Details
    ISSN: 1572-8730
    Keywords: Łukasiewicz algebra ; natural duality ; piggyback duality ; free algebra ; equational base
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Natural dualities are developed for varieties ofn-valued Łukasiewicz algebras with and without negation. These dualities are based on hom-functors, and parallel Stone duality for Boolean algebras. A translation is described which relates the natural dualities to the corresponding restricted Priestley dualities. This enables a unified approach to free algebras to be presented, whence R. Cignoli's characterisations of the finitely generated free algebras are elucidated and new descriptions of arbitrary free algebras obtained. Finally it is shown how dualities for subvarieties encode equational bases. An alternative approach is thereby provided to results of M.E. Adams and R. Cignoli.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01053004
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  • 7
    Electronic Resource
    Electronic Resource
    Finitely generated free modular ortholattices. I (1997)
    Springer
    International journal of theoretical physics 36 (1997), S. 2639-2660 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A description is given of then-generated free algebras in the variety of modular ortholattices generated by an ortholatticeMO 2 of height 2 with 4 atoms. In the subvariety lattice of orthomodular lattices, the varietyV(MO 2) is the unique cover of the variety of Boolean algebras, in whichn-generated free algebras were described by G. Boole in 1854. It is shown that then-generated free algebra in the varietyV(MO 2) is a product of then-generated free Boolean algebra2 2 n and Φ(n) copies of the generatorMO 2, and formula is presented for Φ(n). To achieve this result, algebraic methods of the theory of orthomodular lattices are combined with recently developed methods of natural duality theory for varieties of algebras.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02435704
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  • 8
    Electronic Resource
    Electronic Resource
    Catalytic distributive lattices and compact zero-dimensional topological lattices (1984)
    Springer
    Algebra universalis 19 (1984), S. 322-329 
    Details
    ISSN: 1420-8911
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The (Priestley) dual spaces ofD 01-catalytic lattices are analysed and shown to be precisely the compact zero-dimensional topological lattices. This characterisation is used to prove that a bounded distributive lattice isD 01-catalytic if and only if it is a retract of one freely generated by an ordered set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01201099
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  • 9
    Electronic Resource
    Electronic Resource
    The determination of subvarieties of certain congruence-distributive varieties (1994)
    Springer
    Algebra universalis 32 (1994), S. 44-62 
    Details
    ISSN: 1420-8911
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This note concerns the problem of determining equational bases for the subvarieties of a congruence-distributive variety generated by a finite algebra. It describes a procedure, based on the duality for finite distributive lattices, which converts a local solution to this problem into a global one. For many varieties of distributive-lattice-ordered algebras a local solution of the required form can be obtained from a natural duality in which identities are very conveniently encoded.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01190816
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  • 10
    Electronic Resource
    Electronic Resource
    Endoprimal distributive lattices are endodualisable (1995)
    Springer
    Algebra universalis 34 (1995), S. 444-453 
    Details
    ISSN: 1420-8911
    Keywords: Distributive lattice ; endoprimal algebra ; natural duality ; 06D05 ; 08B99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract L. Márki and R. Pöschel have characterised the endoprimal distributive lattices as those which are not relatively complemented. The theory of natural dualities implies that any finite algebraA on which the endomorphisms of A yield a duality on the quasivariety $$\mathbb{I}\mathbb{S}\mathbb{P}(A)$$ is necessarily endoprimal. This note investigates endodualisability for finite distributive lattices, and shows, in a manner which elucidates Márki and Pöschel's proof, that it is equivalent to endoprimality.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01182100
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