By I. N. Herstein
Starting summary Algebra with the vintage Herstein therapy.
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Additional info for Abstract Algebra (3rd Edition)
Sn , as well as their embeddings as subsorts of each other. Thus the carriers for the sorts si are required to be isomorphic. The si must be distinct. ’ in ASCII. It provides an explicit speciﬁcation of the values of the subsort s of s , in contrast to the implicit speciﬁcation provided by using subsort declarations and overloaded operation symbols. The subsort deﬁnition declares the sort s; it declares the embedding of s as a subsort of s , which must already be declared in the local environment; and it asserts that the values of s are precisely (the projection of) those values of the variable v from s for which the formula F holds.
Sect. 2). ID ::= MIX-TOKEN ::= BRACED-ID ::= BRACKET-ID::= EMPTY-BRS ::= id MIX-TOKEN+ TOKEN | PLACE | BRACED-ID | BRACKET-ID | EMPTY-BRS braced-id ID bracket-id ID empty-braces | empty-brackets An identiﬁer may also be a mixﬁx identiﬁer 4 ‘t0 __ . . __tn ’, consisting of a sequence of possibly-empty mixﬁx tokens ti interspersed with place-holders, each place-holder being written as a pair of underscores ‘__’. Mixﬁx identiﬁers allow the use of mixﬁx notation for application of operations and predicates to argument terms in concrete syntax.
Due not only to overloading of predicate and/or operation symbols, but also to implicit embeddings from subsorts into supersorts, a well-sorted atomic formula may have several non-equivalent expansions, preventing it from being well-formed. Qualiﬁcations on operation and predicate symbols, or explicit sorts on terms, may be used to determine the intended expansion (up to the equivalence indicated above) and make the enclosing formula well-formed. Membership MEMBERSHIP ::= membership TERM SORT A membership formula is written: T ∈s The sign displayed as ‘∈’ is input as ‘in’.
Abstract Algebra (3rd Edition) by I. N. Herstein