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In this survey article, we introduce the systems of semigroups on a semilattice together with some modified Green's relations on abundant semigroups. We demonstrate that the regular ...-cryptogroup can be constructed by using an ...G system of completely e J semigroups. Some other kind of generalized cryptogroups can also be similarly constructed. Thus, the systems of semigroups is a powerful tool for constructing the generalized cryptogroups. Our result concerning the regular ...-cryptogroups extends a well known result of M. Petrich on completely regular semigroups and some other known results of J. B. Fountain on superabundant semigroups. [ABSTRACT FROM AUTHOR]

The theory of congruences is an important topic in semigroup theory. There are plenty of results about congruence theory on regular semigroups. Abundant semi- groups are generalizations of regular semigroups. There have been many attempts to work on congruence theory on abundant semigroups. In this paper, some results about congruences theory on abundant semigroups are summarized. We focus our survey mainly on the work of the Chinese people, of course also related to some of the work overseas. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a fast algorithm which can be used to determine the triangular factorization of an inverse of a symmetric matrix of Loewner type by using O(n²) operations. [ABSTRACT FROM AUTHOR]