Biography

Prof. Jie Fang

Guangdong Polytechnic Normal University, China

In 2001, an open question that was posed by Blyth, Silva and Varlet in [2] is the following: for an Ockham algebra L determine the congruences on L that are kernels of endomorphisms on L. Whereas a general solution to this is stll an open problem, there has been some progress in investigating kernels of endomorphisms in various lattice-ordered algebras. In this con- nection, an algebra A is said to have the endomorphism kernel property if every congruence on A, other than the universal congruence, is a kernel of an endomorphism on A. A strengthening of this notion was given by Blyth and Silva [6] in the context of Ockham algebras. Specically, if A is an algebra and # is a congruence on A then an endomorphism e on A is said to be compatible with # if (8x; y 2 A) (x; y) 2 # ) (e(x); e(y)) 2 #: If such an endomorphism e is compatible with every congruence on A then it is said to be strong. Then A is said to have the strong endomorphism ker- nel property if every congruence on A, other than the universal congruence, is the kernel of a strong endomorphism on A. In [6], Blyth and Silva investigated Ockham algebras with the strong en- domorphism kernel property by way of Priestley duality. They particularly showed that if L is an MS-algebra with dual space X that has the strong endomorphism kernel property then X can be characterized in terms of 1- point compactications of discrete spaces. A similar approach adopted in some lattice-ordered algebras such as distributive p-algebras, distributive double p-algebras and double MS-algebras. By a direct algebraic way, the class of Heyting algebras and class of semilattices with this property also can be characterised. We recall that a Priestley space is a compact totally order-disconnected topological space. For a Priestley space X, we shall denote by O(X) the set of all clopen down-sets of X. Then O(X) is a distributive lattice. Conversely, if L is a distributive lattice, then (X; ;) is a Priestley space, where X = Ip(L) is the lattice of prime ideals of L and the topology has as a base the sets fx 2 Ip(L) j x 3 ag and fx 2 Ip(L) j x 63 ag. These constructions give that L ' O(Ip(L)) and X ' Ip(O(X) ; ;). The power of duality theory is particular evident in the study of con- gruence relations. If L is a distributive lattice and X is the dual space of L then for every closed subset Q of X the relation Q dened on O(X) by (A;B) 2 Q () A \ Q = B \ Q is a congruence. Moreover, the lattice of congruences on L is dually iso- morphic to the lattice of closed subsets of X.

**RESEARCH INTERESTS**

1. Lattice theory and universal algebra, particularly in distributive lattice with unary operations

2. Ordered algebraic structures

**EDUCATION**

University of St. Andrews, United Kingdom (1988-91)

Ph.D. supervised by T. S. Blyth.

South China University of Technology (1984-87)

M.Sc.

The Chinese Academy of Sciences, Beijing (Summer 1986)

Individual study supervised by Binren Li

South China Normal University (1982-83)

Visiting study in the Department of Mathematics

Zhongshan University, China (1978-82)

B.Sc. in Mathematics

THESES

1. Contributions to the theory of Ockham algebras, Ph.D. thesis, University of St Andrews, Scotland, 1991

2. Non-normal operators and their applications, M.Sc. thesis (Mathematics), South China University of Technology, Guangzhou, China, 1987

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