Herberto de Jesus da Silva
@unl.pt
Departamento de Matemática da Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
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
Scopus Publications
Scopus Publications
T. S. Blyth and H. J. Silva
Springer International Publishing
T. S. Blyth and H. J. Silva
Springer Science and Business Media LLC
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.
H. J. Silva
Springer Science and Business Media LLC
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.
T. S. Blyth and H. J. Silva
Informa UK Limited
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.
T. S. Blyth, J. Fang, and H. J. Silva
Informa UK Limited
Abstract 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.
P. J. V. Rodrigues and H. J. Silva
Informa UK Limited
Abstract 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).
T.S. Blyth and H.J. Silva
Springer Science and Business Media LLC
Abstract. 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.
H.J. Silva
Informa UK Limited
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.
T. S. Blyth and H. J. Silva
Springer Science and Business Media LLC
Given an ordered set P and an antitone map g : P → P, we obtain necessary and sufficient conditions for the existence of an odd positive integer k such that gk is isotone. The results obtained have a natural application to the dual space of an Ockham algebra. In particular, we determine the cardinality of the endomorphism semigroup of a finite subdirectly irreducible Ockham algebra.
T.S. Blyth and H.J. Silva
Informa UK Limited
Given any family of normal subgroups of a group, we construct in a natural way a certain monoid, the group of units of which is a semidirect product. We apply this to obtain a description of both the semigroup of endomorphisms and the group of automorphisms of an Ockham algebra of finite boolean type. We also determine when such a monoid is regular, orthodox, or inverse
T. S. Blyth and H. J. Silva
Cambridge University Press (CUP)
AbstractIf (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.
T.S. Blyth and H.J. Silva
Informa UK Limited
If an Ockham algebra L belongs to a Berman class and its endomorphism semigroup End L is regular then necessarily L ∈ Kp 2 for some p. For a given L∈Kp 2 the question of precisely when End L is regular is solved in the case where L is subdirectly irreducible. Using a particular construction, we show that every Berman class Kp 2 contains an algebra L for which End L is an inverse semigroup.
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.