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Jie Fang
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. Speci cally, 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 compacti cations 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 de ned 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.


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

2.     Ordered algebraic structures


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

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


South China University of Technology  (1984-87)



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 



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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