User:Inthar/Style guide: Difference between revisions
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* Zero-indexing is used for indices. | * Zero-indexing is used for indices. | ||
* A ''(linear) word'' is a function <math>w : [n]_0 \to \mathcal{A}</math> where <math>\mathcal{A}</math> is a set of letters and <math>n \in \mathbb{Z}_{>0}.</math> ''n'' is called the ''length'' of ''s''. | * A ''(linear) word'' is a function <math>w : [n]_0 \to \mathcal{A}</math> where <math>\mathcal{A}</math> is a set of letters and <math>n \in \mathbb{Z}_{>0}.</math> ''n'' is called the ''length'' of ''s''. | ||
* A ''based circular word'' is a function <math> s: \mathbb{Z}/n \to \mathcal{A}.</math> A ''period'' of a circular word is the minimal <math>p \ | * A ''based circular word'' is a function <math> s: \mathbb{Z}/n \to \mathcal{A}.</math> A ''period'' of a circular word is the minimal <math>p, 1 \le p \le |s|,</math> such that for all ''i'', <math>s[i+p]=s[i].</math> If the period of ''s'' is equal to the length of ''s'', then ''s'' is called primitive. | ||
* A ''(free) circular word'' is a class of based circular words equivalent under rotation: <math>\{x\mapsto s[x], x\mapsto s[x+1], ..., x\mapsto s[x+|s|-1] \}</math> for ''s'' a based circular word. | * A ''(free) circular word'' is a class of based circular words equivalent under rotation: <math>\{x\mapsto s[x], x\mapsto s[x+1], ..., x\mapsto s[x+|s|-1] \}</math> for ''s'' a based circular word. | ||
* The length of a linear, based circular, or free circular word ''s'' is denoted {{len|''s''}} or len(''s''). | * The length of a linear, based circular, or free circular word ''s'' is denoted {{len|''s''}} or len(''s''). | ||
* Circular words may be treated as (based) infinite words. Thus for ''m'', ''n'' integers, ''m'' < ''n'', then ''s''[''m''] denotes ''s''[''m'' mod {{len|''s''}}]. The notation ''s''[''m'':''n''] denotes the (''n'' − ''m'')-letter word ''s''[''m'']''s''[''m''+1]...''s''[''n''−1], where all indices are taken mod {{len|''s''}}. | * Circular words may be treated as (based) infinite words. Thus for ''m'', ''n'' integers, ''m'' < ''n'', then ''s''[''m''] denotes ''s''[''m'' mod {{len|''s''}}]. The notation ''s''[''m'':''n''] denotes the (''n'' − ''m'')-letter word ''s''[''m'']''s''[''m''+1]...''s''[''n''−1], where all indices are taken mod {{len|''s''}}. | ||
* Substitution: If ''w'' is a linear or based circular word in '''X''' and possibly other letters, and ''u'' is a based circular word in '''b''' and '''c''', then <math>\mathsf{subst}(w, \mathbf{X}, u)</math> denotes the word ''w'' but with the ''i''th occurrence of '''X''' replaced with ''u''[''i''] (for ''i'' ≥ 0). | * Substitution: If ''w'' is a linear or based circular word in '''X''' and possibly other letters, and ''u'' is a based circular word in '''b''' and '''c''', then <math>\mathsf{subst}(w, \mathbf{X}, u)</math> denotes the word ''w'' but with the ''i''th occurrence of '''X''' replaced with ''u''[''i''] (for ''i'' ≥ 0). | ||
== Algebraic structures == | == Algebraic structures == | ||
* <math>\mathrm{JI}\langle p_1, ..., p_r \rangle</math> is the ''p''<sub>1</sub>.[...].''p''<sub>''r''</sub> subgroup, the subgroup of <math>(\mathbb{Q}_{>0}, \cdot)</math> generated by rationals <math>p_1, ..., p_r.</math> | * <math>\mathrm{JI}\langle p_1, ..., p_r \rangle</math> is the ''p''<sub>1</sub>.[...].''p''<sub>''r''</sub> subgroup, the subgroup of <math>(\mathbb{Q}_{>0}, \cdot)</math> generated by rationals <math>p_1, ..., p_r.</math> | ||