Kandasamy W. B. V Smarandache Semirings, Semifiel

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.Then S is a Smarandache commutative congruencesimple finite semiring.Proof: S = Cn × Zp has a S-subsemiring which is a Smarandache-commutativecongruence simple finite semiring.(For more details refer Chris Monico).Remark: Only S-semiring of II level can be S commutative c-simple finite semiring.DEFINITION 4.5.5: Let S be a S-semiring of level II.S is said to satisfy Smarandache assending chain condition II (S-accII) if S1 ⊂ S2 ⊂ … is a monic ascending chain of S-ideals Si (of level II) and there exists a positive integer r such that Sr = Ss for all s ≥ r.Thus we see S-acc II is different from S-acc.The reader to requested to constructexamples in S-semirings of level II which have both sets of S-acc's that is S-acc andS-acc II are satisfied.Clearly a S-semiring of level I can never satisfy S-acc II.Similarly we define S-dcc II for S-semirings of level II.Here also a S-semiring oflevel I can never satisfy the S-dcc II condition.Now we proceed on to define compactsemiring of level II.DEFINITION 4.5.6: Let S be a S-semiring of Level II.S is said to be a Smarandachecompact semiring of level II (S-compact II semiring), if A ⊂ S where A is a S-subsemiring II of S is a compact subsemiring under the operations of S.Here also if Sis a S-semiring of level I then S cannot be a S-compact II semiring.DEFINITION 4.5.7: Let S be a semiring, which is a S-semiring of II level.S is said to be Smarandache ∗ semiring of level II (S-∗ semiring II) if S contains a proper subset A satisfying the following conditions1.A is a subsemiring of S2.A is a S-subsemiring of Level II.3.A is a ∗ -semiring.88So if S is a S-∗ semiring II then S need not be a S ∗ -semiring.DEFINITION 4.5.8: Let S be a S-semiring of level II.S is said to be Smarandacheinductive ∗ semiring II (S inductive ∗ -semiring II) if S contains a proper subset A satisfying the following conditions:1.A is a subsemiring of S.2.A is a S-subsemiring II3.A is a inductive ∗ - semiringIt is easily verified a S-semiring of level I can never be a S-∗ semiring of level II or S-inductive ∗ -semiring of level II.DEFINITION 4.5.9: Let S be a S-semiring of level II.S is said to be Smarandachecontinuous semiring of level II (S-continuous semiring II) if a proper subset A of Ssatisfies the following condition:1.A is a S-subsemiring of level II2.A is a continuous semiringIn case of S semiring of level II also, the definition remains the same as that of thedefinition of S-idempotent semiring given in S-semiring of level I.While defining forS-semiring of level II the concept of Smarandache e-semiring we replace thesubsemiring A which is a S-subsemiring by S-subsemiring of level II and A is a e-semiring.Thus with these definitions about substructures in S-semiring of level II wepropose the following problems for the reader to solve.Notation: A semiring of all types in level two will shortly be denoted by S-semiring II for example S-continuous semiring II etc.PROBLEMS:1.Give an example of a S-semiring of level II ofi.Finiteorder.ii.Infiniteorder.2.Find for the S-semiring, S = Z15 × C5i.S-subsemiringII.ii.S-idealII.3.Can the S-semiring II where S = Z17 × C8 be a S-continuous semiring II?4.Is the S-semiring II where S = Q × C2 be a S-continuous semiring II?5.Is the semiring S = Z12 × G be a S-e semiring II?6.Find a S-semiring II which is a S-compact semiring II.897.Give an example of a S-∗ semiring II.8.Find an example of S-semiring II which is a S-inductive ∗ semiring II.9.Give an example of S-semiring II which is a S-acc II semiring.10.Give an example of a S-semiring II which is ai.S-dcc II semiringii.S-MC II semiringiii.S-mc II semiring.11.Give an example of a S-semiring II which is a S-c-semiring II.12.Can a S-semiring II be not a S-dcc II semiring?4.6 Smarandache Anti SemiringHere we introduce yet an interesting property of semiring viz.Smarandache antisemiring.Florentin Smarandache has introduced a new concept called Smarandacheanti structures.A set that is a strong structure contain a proper subset that has aweaker structure, for example if G is a group, we consider a subset S of G that is asemigroup, for the stronger structure groups contain subsets which are semigroupswhich are known as Smarandache anti semigroups.Suppose Z denotes the groupunder + we see Z+ the set of integers without zero is a semigroup.Like wise weintroduce here the concept of Smarandache anti semiring.DEFINITION 4.6.1: Let R be a ring.R is said to be Smarandache anti semiring (S-anti semiring) if R contains a subset S such that S is just a semiring.Example 4.6.1: Let Z be the ring.Z is an S-anti semiring for Z+ the set of positive integers is a semiring.Example 4.6.2: Let Q be the field of rationals, Q+ is the semiring.So Q is S-anti semiring.Example 4.6.3: Let R be the field of reals, R+ is a semiring so R is a S-anti semiring.Example 4.6.4: C be the field of complex numbers.This has subsets Z+, Q+ and R+ to be semirings.Hence C is a S-anti semiring.All these are examples of commutative rings of infinite order, that is of characteristic0.Now we proceed on to study non-commutative rings with characteristic 0.Example 4.6.5: Let M3×3 = {(aij) / aij ∈ Z; the ring of integers} be the set of all 3 × 3matrices under matrix addition and matrix multiplication.M3×3 is a non-commutativering of characteristic 0.Clearly, M3×3 is a S-anti-semiring as M3×3 = {(aij) / aij ∈ Zo} isa semiring.90Example 4.6.6: Let Q[x] be the polynomial ring.The subset P = {Q+[x] / Q+ is the positive rationals is a semiring}, So Q[x] is a S-anti semiring.Example 4.6.7: Let M be any modular lattice having the following Hasse diagram1ab cdd efFigure 4.6.1S = {1, a, b, f, 0} is a semiring, so can we say M is a S-anti semiring of finite order?Note: A modular lattice can never be a ring.Example 4.6.8: Let P3×3 = {(aij) / aij ∈ M} where M is a lattice having the following Hasse diagram:1acbdegfhFigure 4.6.2Let H = {0, e, a, 1} is a distributive lattice.Let M3×3 = {(aij) / aij ∈ H}.Can we sayM3×3 is a S-anti semiring? From these examples we see a distributive lattice is asemiring and the class of distributive lattices is contained in the class modular latticesare never rings.So the concept of S-anti semirings using distributive lattices cannot bedefined.Here M3×3 is a non-commutative algebraic structure.Now we have the91following open problem.Can we say all rings are S-anti semirings? The answer to this question is no, for when we take the rings Zn = {0, 1, 2, … , n-1}, n any positiveinteger we see Zn has no subset which is a semiring [ Pobierz całość w formacie PDF ]

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