In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as： aγb if and only if there exist positive integers m and n such that b^m∈E （Sa）S an...In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as： aγb if and only if there exist positive integers m and n such that b^m∈E （Sa）S and a^n ∈ （Sb）S for all a and b in S. We have proved that S/γ is a maximal separative Semilattice homomorphic image of S. Every AG-groupoid S is uniquely expressible as a semilattice Y of archimedean AG-groupoids Sa （a∈ Y）. The semilattice Y is isomorphic to S/γ and the Sγ （a ∈ Y） are the equivalence classes of S mod V.展开更多
In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of ...In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of amalgamation of the components of F is also given. Some elements a＋b√3/c of Q（i, √3） and their conjugates a-b√3/c a c over Q（i） have different signs in the orbits of the biquadratic field Q（i, √3） when acted upon by F. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Fa, and they form a closed path which is the only closed path in the orbit Гa. We also devise a procedure to obtain ambiguous numbers of the form a-b√3/c, where b is a positive integer.展开更多
基金Financially supported by Higher Education Commission of Pakistan
文摘In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as： aγb if and only if there exist positive integers m and n such that b^m∈E （Sa）S and a^n ∈ （Sb）S for all a and b in S. We have proved that S/γ is a maximal separative Semilattice homomorphic image of S. Every AG-groupoid S is uniquely expressible as a semilattice Y of archimedean AG-groupoids Sa （a∈ Y）. The semilattice Y is isomorphic to S/γ and the Sγ （a ∈ Y） are the equivalence classes of S mod V.
文摘In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of amalgamation of the components of F is also given. Some elements a＋b√3/c of Q（i, √3） and their conjugates a-b√3/c a c over Q（i） have different signs in the orbits of the biquadratic field Q（i, √3） when acted upon by F. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Fa, and they form a closed path which is the only closed path in the orbit Гa. We also devise a procedure to obtain ambiguous numbers of the form a-b√3/c, where b is a positive integer.
