Footnotes
Keywords: main

... Language1
  This is an incomplete draft reporting work in progress.

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... (SQL).2
  See, e.g., [12] for an excellent, easily accessible tutorial.

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....3
  We call these monoids ``primitive'' following the presentation of Fegaras and Maier [9] as it adheres to a more operational (as opposed to mathematical) approach more suitable to computer-scientists. Mathematically, however, these should be called ``semantic'' monoids since they are interpreted by the computation semantics of their operations.

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... algebra.4
  For a fixed set of base elements and operations (which constitute what is formally called a signature), the syntactic algebra is unique (up to isomorphism). This algebra is also called the free, or the initial, algebra for its signature.

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... structures.5
  Note that this ambiguity never arises for semantic algebras whose operations are interpreted into a unique result.

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... commutativity.6
  Such are important considerations in the field of term rewriting [8], where the problem of finding canonical term representations for equational theories was originally addressed by Donald Knuth and Peter Bendix in a seminal paper proposing a general effective method--the so-called Knuth-Bendix Completion Algorithm [13]. The problem, incidentally, is only semi-decidable. In other words, the Knuth-Bendix algorithm may diverge, although several interesting variations have been proposed for a wide extent of practical uses (see [8] for a good introduction and bibliography).

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... ``curryed.''7
  Recall that a curryed form of an n-ary function f is obtained when f is applied to less arguments than it expects; i.e., , for . In the -calculus, this form is simply interpreted as the abstraction . In their fully curried form, all n-ary functions can be seen as unary functions; indeed, with this interpretation of curried forms, it is clear that .

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... online8
  `http://www.acm.org/sigs/sigmod/record/issues/9403/Comprehension.ps`

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... online9
  `http://www.cs.washington.edu/homes/echris/papers/apl99.ps`

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... online10
  `ftp://ftp.cs.washington.edu/tr/1998/11/UW-CSE-98-11-01.PS.Z`

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... online11
  `http://www-sal.cs.uiuc.edu/~nachum/papers/taste-fixed.ps.gz`

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... online12
  `http://lambda.uta.edu/tods00.ps.gz`

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... online13
  `ftp://ftp.informatik.uni-konstanz.de/pub/dbis/Publications/HS:BTW91. ps.gz`

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... online14
  `http://w3.one.net/~jhoffman/sqltut.htm`

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... online15
  `http://www.cs.toronto.edu/~libkin/papers/sigmod96a.ps.gz`

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... online16
  `file:/nfs/opl/matisse/dev/opl_syntax_critic.html`

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... online17
  `ftp://ftp.cs.wisc.edu/pub/tech-reports/reports/1993/tr1161.ps.Z`

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... online18
  `ftp://ftp.cis.upenn.edu/pub/ircs/tr/94-09.ps.Z`

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