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...loop of a loop L (denoted by A(L)) is the subloop generated by all of its associators, that is 〈{x ∈ L / x = (a, b, c) for some a, b, c ∈ L}〉. DE... ...s which is a distinctly an important property enjoyed only by Smarandache associators. Further these concepts leads us to several open unsolved prob... ...ator subloop) of L is denoted by L A is the subloop generated by all the associators in A, where A is a S-subloop of L ( A ⊂ L) i.e. L A = 〈{x ∈ A/...
...ubloop of a loop L denoted by A(L) is the subloop generated by all of its associators, that is A(L) = 〈{x ∈ L / x = (a, b, c ) for some a, b, c ∈ L ... ...L 1 ∪ L 2 denoted by B(AL) is the {subloop of L 1 generated by all its associators} ∪ {subloop of L 2 generated by all of its associators} = A(... ...biloop P of L denoted by A(L s ) is the S-sub-biloop generated by all the associators, that is, 〈{x ∈ P/ x = (a, b, c) for some a, b, c ∈ P} 〉. If A... ...op P. Here also as in case of S-commutators we see there can be several S-associators unlike a unique associator in case of biloops. DEFINITION ... ...of the biring R denoted by A(R) is the sub-biring generated by all of its associators that is by 〈{x ∈ R = R 1 ∪ R 2 x = (a, b, c) for some a, b,... ... is a S-quad ring. The notions of Smarandache commutators, Smarandache associators, Smarandache center, Smarandache Moufang center, Smarandache n...
...oop of a loop L (denoted by A (L)) is the subloop generated by all of its associators, that is, 〈 {x ∈ L / x = (a, b, c) for some a, b, c ∈ L} 〉. ... ...tor subloop) of L is denoted by L A , is the subloop generated by all the associators in A, where A is a S-subloop of L (A ⊂ L) i.e., L A = 〈{x ∈ A...
...er is increased. As an instance of this, I men tion the following; when the Associators petition was before the House of Assembly of Pennsylvania; tw...
...illiam Allen, Abram Tay lor, Esqr., and myself were sent to New York by the associators, commission’d to borrow some cannon of Governor Clinton. He a... ...r carriages, which we soon transported and mounted on our battery, where the associators kept a nightly guard while the war lasted, and among the rest...