Let F X) be the free algebraic structure of signature (Sigma ) freely generated by X.
The set T is freely generated by V with the shuffle composition.
Since the language of L is freely generated by the union of the signatures of L1 and L2, it contains mixed formulas such as φ = □1(□2p → p).
The set of well-formed formulas of propositional logic are freely generated from the set of atoms (i.e., atomic sentences) by the basic logic operations.
During this analysis, the data freely generated the postulated contend-related structure model, together with the predicted parameters of the model.
The key notion is that of a graded algebra: a freely generated algebra (W,·,1, ⎵) where the monoid (W,·,1) of the interpretation of L* is augmented with a distinguished generator ⎵ called the separator.
Given the universal freely generated syntactic calculus discussed above, the task of providing a categorial grammar for a particular language is reduced to specifying its lexicon: the typelogical framework, in this sense, represents the culminating point of 'lexicalizing' grammar formalisms.
Consider the set of symbols ({z({vec i, u} mid vec iin {1,2,3}^k, uin Wcup W_0}) and the multiplicative Abelian group (T_1) of exponent p freely generated by this set.
From the semantic side, a C-algebra B together with a subset X of B thought of as variables is said to be a free C-algebra on X, or is freely generated by X, when, given any C-algebra A, any valuation in A of the variables in X (that is, any function f : X→A) uniquely extends to a homomorphism h: B→A.
It was found out in the early 1990s that the part on lattice theory contains the solution of a decision problem, called the word problem for freely generated lattices, the known solution of which stemmed from 1988! Skolem's terminology and notation in lattice theory are those of Schröder and that is part of the reason why his work was a lost opportunity for proof theory.
