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.In viewof this we have the following theorem, which gives a class of S-commutative c-simplefinite semirings.THEOREM 4.4.2: V(S(n)) is a S-commutative congruence simple semiring for all finite positive integer n.Proof: Since every S(n) is a S-semigroup and every S(n) has proper subsets which areabelian groups we have the result to be true.THEOREM 4.4.3: V(S(n)) has at least (n-1) proper subsets which are abelian groups.So V(S(n)) is a S-commutative congruence simple finite semiring.Proof: Obvious from the fact for every m, 1 < m < n we have cyclic group of order m, which we denote by Gm, so that V(Gm) is a commutative, congruence simple finitesemiring, hence V(S(n)) is a S-commutative congruence simple finite semiring.Using the classical theorem of Cayley which states, "Every group is isomorphic to asubgroup of A(S) for some appropriate S".We can modify it and restate it as " Every finite abelian group is isomorphic to a subgroup of Sn for some appropriate n".Now81we use the analogue of this classical theorem for S-semigroups which is given by us as "Every S-semigroup is isomorphic to a S-subsemigroup in S(n)".Using this we have the following very interesting result.THEOREM 4.4.4: Let G be a finite abelian group V(G) be isomorphic to S for somecommutative congruence simple finite semiring.Then there exists a suitable n so thatevery V(G) has a isomorphic image in the S-c-simple semiring V(S(n)) for a suitablen.Proof: We know by classical Cayley's theorem.G is isomorphic a subgroup in Sn for asuitable n.Now each Sn ⊂ S(n) and we have S(n) is a S-semigroup.Now V(G) ⊂ V(Sn) s⊂ V(S(n)) so we have the theorem to be true.Example 4.4.3: Let Cn be a chain lattice with n-elements, Mt×t be the collection of all t × t matrices with entries from Cn.Mt×t is a S-finite congruence simple semiring.Thus we have a class of Smarandache finite congruence simple semiring for varying t.DEFINITION 4.4.3: Let S be a semiring we say S is a Smarandache right chainsemiring (S-right chain semiring) if the S-right ideals of S are totally ordered byinclusion.Similarly we define Smarandache left chain semiring (S-left chainsemiring).If all the S-ideals are totally ordered by inclusion we say S is aSmarandache chain semiring (S-chain semiring).Example 4.4.4: Let C9 be a chain lattice, which is a semiring of order 9.Clearly C9 is a S-chain semiring.THEOREM 4.4.5: All chain lattices are S-chain semirings.Proof: Left for the reader to verify.DEFINITION 4.4.4: Let S be a semiring.If S1 ⊂ S2 ⊂ … is a monotonic ascending chain of S-ideals Si, and there exists a positive integer r such that Sr = Ss for all s ≥ r.then we say the semiring S satisfies the Smarandache ascending chain conditions (S-acc)for S-ideals in the semiring S.Example 4.4.5: If we take the semiring S to be any chain lattice S, then it satisfies S-acc.DEFINITION 4.4.5: Let S be any semiring.We say S satisfies Smarandache descendingchain condition (S-dcc) on S-ideals Si if every strictly decreasing sequence of S-idealsN1 ⊃ N2 ⊃ N3 ⊃ … in S is of finite length.The Smarandache min condition (S – mc) for S-ideals holds in S if given any set P of S-ideals in S, there is an ideal of P that doesnot properly contain any other ideal in the set P.DEFINITION 4.4.6: Let S be a chain ring such that S-acc for ideals holds in S.The Smarandache maximum condition (S-MC) for S-ideals holds in S if for every non-empty set P of S-ideals in S contains a S-ideal not properly contained in any other S-ideal of the set P.82DEFINITION 4.4.7: Let S be a semiring we say S is a Smarandache compact semiring(S-compact semiring) if A ⊂ S where A is a S subsemiring of S is a compact semiring under the operations of S.It is important to note if S is to be a S-compact semiring itis not necessary that S is a compact semiring.If S has a S-subsemiring which is acompact subsemiring then it is sufficient.So we have the following.THEOREM 4.4.6: If S is a compact semiring and S is a S-semiring then S is a S-compact semiring provided S has a S-subsemiring.Proof: Obvious by the very definition of S-compact semiring.Example 4.4.6: All chain semirings Cn[x] are S-compact semirings.For Cn ⊂ Cn[x] is a S-subsemiring of Cn[x] which is a compact semiring as a.b = b and a + b = a.Example 4.4.7: Let X be a set with n elements, P(X) the power set of X which is the Boolean algebra.Hence P(X) is a semiring.P(X) is a S-compact semiring for everychain in P(X) connecting X and φ is a S-subsemiring which is a compact semiring.DEFINITION 4.4.8: Let S be a semiring, S is said to be Smarandache ∗ -semiring (S-∗ -semiring) if S contains a proper subset A satisfying the following conditions:1.A is a subsemiring2.A is a S-subsemiring3.A is a∗ -semiringSo if S is a ∗ - semiring and if S has a S-subsemiring then obviously S is a S-∗ -semiring.DEFINITION 4.4.9: Let S be any semiring.We say S is a Smarandache inductive ∗ -semiring (S-inductive ∗ -semiring) if S contains a proper subset A such that thefollowing conditions are true1.A is a subsemiring of S.2.A is a S-subsemiring of S3.A is inductive ∗ -semiring.THEOREM 4.4.7: If S is a inductive ∗ -semiring and S has a S-subsemiring then S is a Smarandache inductive ∗ -semiring.Proof: By the very definition of S-inductive ∗-semiringsDEFINITION 4.4.10: A semiring S is said to be a Smarandache continuous semiring(S-continuous) if a proper subsemiring A of S satisfies the following 2 conditions1.A is a S- subsemiring.2.A is a continuous semiring.83DEFINITION 4.4.11: Let S be a semiring.S is said to be a Smarandache idempotentsemiring (S-idempotent semiring) if A proper subset P of S, which is a subsemiring ofS satisfies the following conditions:1.P is a S-subsemiring.2.P is an idempotent semiring.Example 4.4.8: Let C7[x] be the polynomial semiring.C7[x] is a S-idempotentsemiring for C7 ⊂ C7[ x] is1.Subsemiring of C7[x].2.C7 is a S-subsemiring.3.In C7 we have a + a = a for all a ∈ C7.DEFINITION 4.4.12: Let S be a semiring.S is said to be a Smarandache e-semiring (S-e-semiring) if S contains a proper subset A satisfying the following conditions:1.A is a subsemiring2.A is a S-subsemiring3.A is a e-semiringExample 4.4.9: Let Cn[x] be a polynomial semiring over the semiring Cn (the chain lattice with n elements).Cn[x] is a S-e-semiring for Cn ⊂ Cn[x] satisfies all conditions.Cn is clearly a S-e-semiring.DEFINITION 4.4.13: Let S be any semiring.G be a Smarandache semigroup.Considerthe semigroup semiring SG.We call SG the Smarandache group semiring.(S-groupsemiring)Example 4.4.10: Let ZoS(n) be the semigroup semiring.ZoS(n) is a Smarandachegroup semiring.Then the natural question would be what is the definition of Smarandache semigroupsemiring.DEFINITION 4.4.14: Let G be a group, G is a Smarandache anti group, that is Gcontains a proper subset which is a semigroup
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