Study of Algebraic and Topological Aspects of Multiplicative Lattices

In 1854, George Boole (1815–1864) introduced an important class of algebraic structures in his research work on mathematical logic. In his honor these structures have been called Boolean algebras. These are special type of lattices. In particular, congruence lattices play an important role. It was E.Schroder, who about 1890, considered the lattice concept in today‘s sense. At approximately the same time, R.Dedikind developed a similar concept in his work on groups and ideals. Dedikind defined modular and distributive lattices which are types of lattices of that are important in applications. The rapid development of lattice theory started around 1930. We could say that Boolean lattices or Boolean algebras are the simplest and at the same time the most important lattices for applications. It was Garrett Birkhoff‘s work in the mid-thirties that started the general development of lattice theory. In a Brilliant series of papers he demonstrated the importance of the lattice theory and showed that it ii provides a unifying framework for previously unrelated developments in any mathematical disciplines. During a pivot step, we make the value of a nonbasic variable just large enough to get the value of a basic variable down to zero. This, however, might never happen. If we now try to bring x2 into the basis by increasing its value, we notice that none of the tableau equations puts a limit on the increment. We can make x2 and z arbitrarily large the problem is unbounded. By letting x2 go to infinity we get a feasible halfline - starting from the current BFS - as a witness for the unboundedness. In our case this is the set of feasible solutions with initial table an After one pivot step with x1 entering the basis we get the tableau :

happens in the degenerate case: some tableau equation limits the increment to zero so that no progress in z is possible. Study of algebraic and topological aspects of multiplicative lattices The only candidate for entering the basis is x2, but the first tableau equation shows that its value cannot be increased without making x3 negative. This may happen whenever in a BFS some basic variables assume zero value, and such a situation is called degenerate. Unfortunately, the impossibility of making progress in this case does not imply optimality, so we have to perform a `zero progress' pivot step. In our example, bringing x2 into the basis results in another degenerate tableau with the same BFS. Nevertheless, the situation has improved. The nonbasic variable x1 can be increased now, and by entering it into the basis, we already obtain the final tableau With optimal BFS x = (x1 ……..x4) = (2,2,0,0) In this example, after one degenerate pivot we were able to make progress again. In general, there might be longer runs of degenerate pivots. Even worse, it may happen that a tableau repeats itself during a sequence of degenerate pivots, so the algorithm can go through an infinite sequence of tableaus without ever making progress. This phenomenon is known as cycling, and an example can be found. If the algorithm does not terminate, it must cycle. This follows from the fact that there are only finitely many different tableaus. and assume there is another tableau T 0 with the same basic and nonbasic variables, i.e. T is the system must hold for all d-vectors xN, and this implies Hence There are two standard ways to avoid cycling: • Bland's smallest subscript rule: If there is more than one candidate xk for entering the basis or more than one candidate for leaving the basis, which is another manifestation of degeneracy, choose the one with smallest subscript k. • Avoid degeneracies altogether by symbolic perturbation. By Bland's rule, there is always a way of escaping from a sequence of degenerate pivots. For this, however, one has to give up the freedom of choosing the entering variable. For us it will be crucial not to restrict the choice of the entering variable, so we will abandon Bland's rule and instead resort to the method of symbolic perturbation, although this requires more computational effort. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis. While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations was an achievement of the twentieth century.

equation . The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if (x, y) is a solution, then the other solutions have the form (x + kv, y − ku), where k is an arbitrary integer, and u and v are the quotients of a and b (respectively) by the greatest common divisor of a and b. Proof: If d is this greatest common divisor, Bézout's identity asserts the existence of integers e and f such that ae + bf = d. If c is a multiple of d, then c = dh for some integer h, and (eh, fh) is a solution. On the other hand, for every pair of integers x and y, the greatest common divisor d of a and b divides ax + by. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution (x, y), we have a(x + kv) + b(y − ku) = ax + by + k(av − bu) = ax + by + k(udv − vdu) = ax + by, showing that (x + kv, y − ku) is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u(x2 − x1) + v(y2 − y1) = 0. As u and v are coprime, Euclid's lemma shows that there exists an integer k such that and . Therefore, and , which completes the proof.

One of the few general approaches is through the Hasse principle. Infinite descent is the traditional method, and has been pushed a long way. The depth of the study of general Diophantine equations is shown by the characterization of Diophantine sets as equivalently described as recursively enumerable. In other words, the general problem of Diophantine analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms. still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds. [4] The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was solved by Andrew Wiles but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated. Other major results, such as Faltings' theorem, have disposed of old conjectures. An example of an infinite diophantine equation is: which can be expressed as "How many ways can a given integer n be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each n forms an integer sequence. Infinite Diophantine equations are related to theta functions and infinite dimensional lattices. This equation always has a solution for any positive n. Compare this to: which does not always have a solution for positive n.

If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation. Examples include the Ramanujan–Nagell equation, , and the equation of the Fermat-Catalan conjecture and Beal's conjecture, with inequality restrictions on the exponents. A general theory for such equations is not available; particular cases such as Catalan's conjecture have been tackled. However, the majority are solved via ad hoc methods such as Stormer's theorem or even trial and error.

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Assistant Professor, Yaduvanshi Degree College, Mahendergarh