An Overview of Boolean Algebras and Pre A* - Algebras

Although each solid Boolean algebra is a Boolean algebra, few out of every odd Boolean algebra need be concrete. Give n a chance to be a without square positive whole number, one not separable by the square of a number, for example 30 yet not 12. The operations of greatest normal divisor, least basic numerous, and division into n (that is, ¬x = n/x), can be appeared to satisfy all the Boolean laws when their arguments range over the positive divisors of n. Subsequently those divisors structure a Boolean algebra. These divisors are not subsets of a set, making the divisors of n a Boolean algebra that isn't concrete according to our definitions. In any case, on the off chance that we represent each divisor of n by the arrangement of its prime factors, we find that this nonconcrete Boolean algebra is isomorphic to the solid Boolean algebra comprising of all arrangements of prime factors of n, with association comparing to least regular various, crossing point to greatest normal divisor, and supplement to division into n. So this example while not technically concrete is at least "morally" concrete via this representation, called an isomorphism. This example is an instance of the accompanying idea.

A Boolean algebra is called representable when it is isomorphic to a solid Boolean algebra. The conspicuous next inquiry is answered decidedly as pursues. Each Boolean algebra is representable. That is, up to isomorphism, abstract and cement Boolean algebras are the same thing. This very nontrivial result relies upon the Boolean prime ideal hypothesis, a decision guideline somewhat weaker than the axiom of decision, and is treated in more detail in the article Stone's representation hypothesis for Boolean algebras. This strong relationship recommends a more fragile result reinforcing the perception in the past subsection to the going with simple consequence of representability. The laws fulfilled by every Boolean polynomial math match with those fulfilled by the prototypical Boolean variable based math. It is flimsier as in it doesn't of itself suggest representability. Boolean algebras are extraordinary here, for instance a connection variable based math is a Boolean polynomial math with extra structure however it isn't the situation that each connection polynomial math is representable in the sense fitting to connection algebras.

answer is "all Boolean laws," which can be characterized as all conditions that hold for the Boolean polynomial math of 0 and 1. Since there are infinitely numerous such laws this is definitely not a horribly satisfactory answer by and by, prompting the following inquiry: does it get the job done to require just finitely numerous laws to hold? On account of Boolean algebras the appropriate response is yes. Specifically the finitely numerous conditions we have recorded above do the trick. We state that Boolean variable based math is

Could this once-over be made shorter yet? Again the appropriate response is yes. In any case, a portion of the above laws are induced by a portion of the others. A satisfactory subset of the above laws contains the sets of associativity, commutativity, and absorption laws, distributivity of ∧ over ∨ (or the other distributivity law—one takes care of business), and the two enhancement laws. Truth be told this is the conventional axiomatization of Boolean polynomial math as an enhanced distributive grid. By presenting extra laws not recorded above it ends up conceivable to shorten the rundown yet further. In 1933, Edward Huntington demonstrated that if the essential tasks are taken to be x∨yand ¬x, with x∧y considered a derived operation (e.g. via De Morgan's law in the form x∧y = ¬(¬x∨¬y)), then the equation ¬(¬x∨¬y)∨¬(¬x∨y) = x along with the two equations expressing associativity and commutativity of ∨ completely axiomatized Boolean algebra. When the only basic operation is the binary NAND operation ¬(x∧y), Stephen Wolfram has proposed in his book A New Kind of Science the single axiom ((xy)z)(x((xz)x)) = z as a one-equation axiomatization of Boolean algebra, where for convenience here xy denotes the NAND rather than the AND of x and y.

A solid Boolean algebra or field of sets is any nonempty set of subsets of a given set X shut under the set operations of association, crossing point, and supplement relative to X.[4] (As an aside, historically X itself was required to be nonempty as well to avoid the degenerate or one-component Boolean algebra, which is the one special case to the standard that all Boolean algebras satisfy the same equations since the degenerate algebra satisfies each equation. Anyway this prohibition clashes with the preferred simply equational meaning of "Boolean algebra," there being no real way to degenerate Boolean algebra and let X be vacant.)

The power set 2X of X, comprising of all subsets of X. Here X may be any set: unfilled, limited, boundless, or even uncountable.

The unfilled set and X. This two-component algebra demonstrates that a solid Boolean algebra can be limited notwithstanding when it comprises of subsets of an unending set. It very well may be seen that each field of subsets of X must contain the unfilled set and X. Thus no smaller example is conceivable, other than the degenerate algebra obtained by taking X to be vacant in order to make the unfilled set and X correspond.

The arrangement of limited and cofinite sets of whole numbers, where a cofinite set is one discarding just limitedly many numbers. This is clearly shut under supplement, and is shut under association because the association of a cofinite set with any set is cofinite, while the association of two limited sets is limited. Convergence behaves like association with "limited" and "cofinite" interchanged.

For a less trivial example of the point made by Example 2, consider a Venn diagram framed by n shut bends partitioning the diagram into 2n districts, and let X be the

In the decimal framework when a number is perused from directly to left every digit is increased by a dynamically higher intensity of 10. These are regularly alluded to as ones, tens, hundreds, thousands, etc. In the twofold framework a similar idea applies with the exception of that the number being raised are forces of two, the digits accordingly speaks to ones, twos, fours, eights, etc. In this tallying framework the main numbers that are utilized are ones. Example of the comparison of Binary and Decimal Systems. Decimal system Binary System

utilizes indistinguishable tasks from the decimal framework however might be viewed as more straightforward. The expansion includes gathering things in gatherings of twos with conveying to the following higher power. carrying to the next higher power.

Binary to Decimal conversion: In a binary number each position corresponds to a power of two. Multiplication is also a straight forward procedure, since each digit is either 0 Or 1; therefore each potential product is either zero or one.

The initial phase in the plan of a circuit is to build up a reality table that demonstrates the yield for every single imaginable information.

Computer performs among other things, all kinds of arithmetic operations. The most basic operation is the addition of two binary digits, which consists of 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 and 1 + 0 = 1. The first three operations had sum that is a single digit, but 1 + 1 has a sum of two digits. The higher bit of this sum is called a carry. In adding two multiple digits numbers a carry is to be added to the next higher digit. The circuit that performs the addition of two bits is called a half adder. The circuit that performs the addition of three bits is called a full adder. A half adder needs two binary inputs (A and B) and two binary out puts ( S = sum; C= carry). Truth Table for Half adder. The fourth row shows 1 + 1 = 10, here 1 is the carry to the next higher power of two.

This thought can be connected to voltages that are available in a physical circuit. There are high voltages and low voltages. High voltage implies that current is streaming, low voltage connotes that there

let high voltage = 1 or let high = 0 low voltage = 0 low = 1 from these we can write truth tables to show the desired operations (or, and). or operation In conclusion if the dualities of high and low, on and off are given values from the elements of the set of boolean algebra many physical electrical problems can be solved.

BOOLEAN ALGEBRA:

Definition: A Boolean algebra is an algebra (B, Ú, Ù, (- )~, 0, 1) with two binary operations, one unary operation (called complementation), and two nullary operations which satisfies : is a distributive lattice; We can easily prove that B.

= {0, 1} is given by Whereis a 2 component lattice with 0 < 1 and where 0¢ = 1, 1¢ = 0.Alternative frameworks of postulates for Boolean Algebras were seriously examined amid the decades 1900 - 1940. E.V. Huntington composed an influential early paper [3] regarding this matter. No attempt will be made here to overview the broad literature on such postulate frameworks. We present here Huntington's postulates in 1.1.3

Let B have one binary operationÚand oneunary operation (-)¢ and define (i)for all a,. Suppose for all a, b, c Î B, (ii); (iii)(iv) = a. Then B is a Boolean algebra.

Let B have one binary operationÙand one unary operation (-)¢and define (i) for all B. Suppose for a, b, c Î B, (ii) (iii) (iv)= a. Then B is a Boolean algebra. Pre A* - Algebra: Definition: An algebra (A,Ú,Ù, (-)~) satisfying: (a) (b) (c) (d) (e) (f) (g) is called a Pre A* - algebra.

If (mn) is an axiom in Pre A* - algebra, then (mn)~is its dual.

algebra

So as to characterize a Boolean algebra, we need the additional idea of complementation. Definition: Complemented Lattice. Let L;, contains a least component, 0, and a greatest component, 1. L; is called a supplemented lattice if and if for each component a œ L, there exists a component an in L with the end goal that an a = 0 and an a = 1. Such a component an is called a supplement of the component a.

1) Venkateswara Rao J. and Praroopa Y. (2010) ―Lattice in Pre A*-Algebra‖, Asian Journal of Algebra, ISSN 1994-540X Volume 4, Number 1, pp. 1-11. 2) Venkateswara Rao J. and Praroopa Y. (2011). ―Pre A*-Algebras and Rings‖, International Journal of Computational Science and Mathematics. ISSN 0974- 3189 Volume 3, Number 2 (2009), pp. 161-172. 3) Venkateswara Rao J. and Praroopa Y. (2011). ―Homomorphisms, Ideals and Congruence Relations onPre A*-Algebra‖, Global Journal of Mathematical Sciences: Theory and Practical. ISSN No 0974 – 3200 Volume 3, Number 2 (2011), pp. 111- 125. 4) Venkateswara Rao J. and Praroopa Y. (2011). ―Logic circuits and Gates in Pre A*- Algebra‖, Asian Journal of Applied Sciences, 4(1): pp. 89-96 5) Venkateswara Rao J. and Praroopa Y. (2011). ―Pre A*-lattices‖, Journal of Algebra, 2011, communicated. 6) Venkateswara Rao J. and Srinivasa Rao K. (2009). Congruence relation on Pre A*- algebra, Journal of Mathematical Sciences, ISSN 0973-4597, Vol.4,Issue 4, November 2009, pp. 295-312. 7) J. U. Ahmed, A. A. S. Awwal (1992). ―Polarization encoded optical shadow-casting arithmetic-logic-unit design: separate and Optics, 31(26), pp. 5622-5631. 8) T. Main, R. J. Feuerstein, H. F. Jordan, V. P. Heuring, J. Feehrer, C. E. Love (1994). ―Implementation of a general-purpose stored-program digital optical computer,‖ Applied Optics, 33(8), pp. 1619-1628. 9) B. Lu, Y. C. Lu, J. Cheng, M. J. Hafich, J. Klem, J. C. Zolper (1996). ―High speed, cascaded optical logic operations using programmable optical logic gate arrays,‖ IEEE Photonics Technology Letters, 8(1), pp. 166-168. 10) B. Wang, F. Yu, X. Liu, P. Gu, J. Tang (1996). ―Optical modified signed-digit addition module based on Boolean polarization-encoded logic algebra,‖ Optical Engineering, 35(10), 2989-2994.

Research Scholar of OPJS University, Churu, Rajasthan