Herberto de Jesus da Silva

@unl.pt

Departamento de Matemática da Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa

Herberto de Jesus da Silva


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https://researchid.co/herbertosilva

EDUCATION

Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia
1996 | Doutoramento em Matemática

Universidade de Lisboa
1987 | Licenciatura em Matemática

RESEARCH INTERESTS

Álgebras de Ockham
Reticulados Distributivos
Álgebra Universal
14

Scopus Publications

Scopus Publications

  • Ockham Algebras—An Urquhart Legacy
    T. S. Blyth, H. J. Silva
    Outstanding Contributions to Logic, 2022
  • Congruence kernels in Ockham algebras
    T. S. Blyth, H. J. Silva
    Algebra Universalis, 2017
    We consider, in the context of an Ockham algebra $${{\\mathcal{L} = (L; f)}}$$L=(L;f), the ideals I of L that are kernels of congruences on $${\\mathcal{L}}$$L. We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel $${I \\neq L}$$I≠L is the intersection of the prime ideals P such that $${I \\subseteq P}$$I⊆P, $${P \\cap f(I) = \\emptyset}$$P∩f(I)=∅, and $${f^{2}(I) \\subseteq P}$$f2(I)⊆P. The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted.
  • Minimal Ockham algebras
    H. J. Silva
    Algebra Universalis, 2012
    An Ockham algebra (L; f) will be called minimal if it has exactly two subalgebras. In the generalised variety Kw, we describe the minimal Ockham algebras.
  • The strong endomorphism kernel property in Ockham algebras
    T. S. Blyth, H. J. Silva
    Communications in Algebra, 2008
    An endomorphism on an algebra 𝒜 is said to be “strong” if it is compatible with every congruence on 𝒜; and 𝒜 is said to have the “strong endomorphism kernel property” if every congruence on 𝒜, different from the universal congruence, is the kernel of a strong endomorphism on 𝒜. Here we consider this property in the context of Ockham algebras. In particular, for those MS-algebras that have this property we describe the structure of their dual space in terms of 1-point compactifications of discrete spaces.
  • The endomorphism kernel property in finite distributive lattices and de Morgan algebras
    T. S. Blyth, J. Fang, H. J. Silva
    Communications in Algebra, 2004
    An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.
  • Direct decompositions of Ockham algebras
    Algebra Colloquium, 2004
  • Ockham Congruences Whose Quotient Algebras Are Boolean
    P. J. V. Rodrigues, H. J. Silva
    Communications in Algebra, 2003
    Given any Ockham algebra, we describe the congruences such that the quotient algebras are boolean. This description is obtained using certain ideals that we call pro-boolean ideals. We prove that every proper pro-boolean ideal is the intersection of a family of falsity ideals. We also determine when every proper pro-boolean ideal is a unique intersection of such ideals. Finally, we show that if an Ockham algebra in the Urquhart class P n+2,n is fixed point free then the corresponding dual space has a fixed point. This result is a natural generalisation of a well known theorem (Blyth, T. S., Varlet, J. C. (1994). Ockham Algebras. Oxford University Press, Theorem 6.3).
  • Ockham algebras arising from monoids
    Algebra Colloquium, 2001
  • Subdirectly irreducible Ockham chains
    T.S. Blyth, H.J. Silva
    Algebra Universalis, 2000
    We show that every subdirectly irreducible Ockham chain belongs to the generalised variety Kω and is countable. Consideration of three particular types of finite Ockham chains, together with their order duals, leads to a determination of the structure of all finite subdirectly irreducible Ockham chains. These belong necessarily to the Berman classes K1,q and we show that there are precisely 6q+2 such chains in K1,q. We also show that there are precisely 14 subdirectly irreducible Ockham chains whose endomorphism semigroup is regular, such chains having at most 5 elements.
  • A general construction of Ockham algebras
    H.J. Silva
    Communications in Algebra, 1999
    Using certain families of filters of a bounded distributive lattice, we provide a general method of constructing Ockham algebras. In particular, by using prime filters, we obtain a method of constructing Ockham algebras (L;f) such that f(L;f) is a chain.
  • Singular Antitone Systems
    T. S. Blyth, H. J. Silva
    Order, 1998
  • Semigroups arising from families of normal subgroups
    T.S. Blyth, H.J. Silva
    Communications in Algebra, 1997
  • Endomorphism regular Ockham algebras of finite boolean type
    T. S. Blyth, H. J. Silva
    Glasgow Mathematical Journal, 1997
  • On Ockham algebras whose endomorphism semigroups are regular
    T.S. Blyth, H.J. Silva
    Communications in Algebra, 1996

RESEARCH OUTPUTS (PATENTS, SOFTWARE, PUBLICATIONS, PRODUCTS)

Congruence kernels in Ockham algebras
Silva, H. D. J. D. & Blyth, T. S., May 2017, In : Algebra Universalis. 78, 1, p. 55-65.

Minimal Ockham algebras
Silva, H. D. J. D., 1 Jan 2012, In : Algebra Universalis. 67, 4, p. 393-395.

The strong endomorphism kernel property in Ockham algebras
Silva, H. D. J. D., 1 Jan 2008, In : Communications in Algebra. 36, 5, p. 1682-1694.

Ockham congruences whose quotient algebras are boolean
Silva, H. D. J. D., 31 Aug 2006, In : Communications in Algebra. 31, 11, p. 5391-5404.

Direct decompositions of Ockham algebras
Silva, H. D. J. D., 1 Jan 2004, In : Algebra Colloquium. 11, 2, p. 239-248.

The endomorphism kernel property in finite distributive lattices and de Morgan algebras
Silva, H. D. J. D., 1 Jan 2004, In : Communications in Algebra. 32, 6, p. 2225-2242.

Ockham algebras arising from monoids
Silva, H. D. J. D., 1 Jan 2001, In : Algebra Colloquium. 8, 3, p. 315-326.

On the endomorphism monoid of a finite subdirectly irreducible Ockham algebra
Silva, H. D. J. D., 1 Jan 2001, UNSOLVED PROBLEMS ON MATHEMATICS FOR THE 21ST CENTURY. Abe, JM. & Tanaka, S. (eds.). NIEUWE HEMWEG 6B, 1013 BG AMSTERDAM, NETHERLANDS: I O S PRESS, p. 149-157.


Subdirectly irreducible Ockham chains
Silva, H. D. J. D., 1 Jan 2000, In : Algebra Universalis. 44, 1-2, p. 1-14.

A general construction of Ockham algebras
Silva, H. J., 1 Jan 1999, In : Communications in Algebra. 27, 9, p. 4561-4567 7p.

Singular Antitone Systems
Blyth, T. S. & Silva, H. J., 1 Jan 1998, In : Order. 15, 3, p. 261-270 10 p.

Endomorphism regular Ockham algebras of finite boolean type
Blyth, T. S. & Silva, H. J., 1 Dec 1997, In : Glasgow Mathematical Journal. 39, 1, p. 99-110 12 p.

Semigroups arising from families of normal subgroups
Blyth, T. S. & Silva, H. J., 1 Jan 1997, In : Communications in Algebra. 25, 3, p. 943-954 12 p.

On Ockham algebras whose endomorphism semigroups are regular
Blyth, T. S. & Silva, H. J., 1 Jan 1996, In : Communications in Algebra. 24, 3, p. 919-928 10 p.