| The origin of concept lattice can be traced back to the 1850, and many scholars have discussed extensively for the related theory of lattice. In 1940,<Lattice Theory> is the advent of the lattice theory. Since then, the features of the lattice with the order structure and the algebraic structure have gradually entered the views of scholars. In 1967, R.Padmanabhan put forward a simplified form of the definition of the algebra lattice, in other words, using two equations instead of eight equations. In 1970, R.N.McKenzie proposed to define algebraic lattice with one equation and proved it. Thus, the algebraic definition, partial definitions of lattice and the preliminary results of the study have been recognized by the academic community.In 1978, George Gratzer analyzed the general theory of lattices in <General Lattice Theory>, which described two kinds of definitions of the lattice as basic lattice theory. With the research on the lattice, we will find that the order structure and the algebra structure of the lattice have different performances. George Gratzer not only came up with the definitions of the algebra lattice and the structure lattice, but also first proved that the order definition of the lattice and the algebraic definition of lattice are equivalent. It shows the variety of the lattice structure. This is a brand new way in lattice theory study on the lattice structure.This paper mainly studies the similarities and differences of the order structure and the algebraic structure of the lattice, and discusses the equivalence of the order lattice and the algebraic lattice, and the equivalence of the semilattice, the sublattice, the homomorphism, the isomorphism, the ideal and the direct product under the order structure and the algebraic structure. We obtain a series of related conclusions. At the same time, the article also describes the performance of the ideal and other related definitions of the lattice under the order structure and the algebraic structure. |
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