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• W1452360569 abstract "This dissertation consists of two parts. Part I examines certain Burnside-type conditions on the multiplicative semigroup of an (associative unital) algebra A. A semigroup S is called n-collapsing if, for every a1, . . . , an 2 S , there exist functions f , g on the set {1, 2, . . . , n} such that s f (1) · · · s f (n) = sg(1) · · · sg(n). If f and g can be chosen independently of the choice of s1, . . . , sn, then S satisfies a semigroup identity. A semigroup S is called n-rewritable if f and g can be taken to be permutations. Semple and Shalev extended Zelmanov’s Fields Medal writing solution of the Restricted Burnside Problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. The primary result of Part I is that the following conditions are equivalent for every algebra A over an infinite field: the multiplicative semigroup of A is collapsing, A satisfies a multiplicative semigroup identity, and A satisfies an Engel identity: [x,m y] = 0. Furthermore, in this case, A is locally (upper) Lie nilpotent. It is also shown that, if the multiplicative semigroup of A is rewritable, then A must be commutative. In Part II of this dissertation, we study algebraic analogues to well-known problems of Philip Hall on the verbal and marginal subgroups of a group. We begin by proving two algebraic analogues of the Schur-Baer-Hall Theorem: if G is a group such that G/Zn(G) is finite, whereZn(G) is the nth higher centre of G, then the (n + 1)st term, n+1(G), of the lower central series of G is also finite; conversely, if n+1(G) is finite, then so is G/Z2n(G). Next, we prove results of a more general type. Given an algebra A and a polynomial f , we define the verbal subspace, SA( f ), of A to be spanned by the set of f -values in A, the verbal subalgebra,AA( f ), and the verbal ideal, IA( f ), of A to be generated by the set of f -values in A. We also define the marginal subspace b SA( f ) of A to be the set of all elements z 2 A such that" @default.
• W1452360569 created "2016-06-24" @default.
• W1452360569 creator A5039941932 @default.
• W1452360569 date "2015-01-01" @default.
• W1452360569 modified "2023-09-24" @default.
• W1452360569 title "Combinatorial Polynomial Identity Theory" @default.
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