Appendix I

Appendix I . Terms and notions

1....5,   1...18
numbering of order 4 and 5 Ls's which we preferred for representing the  structure types in the corresponding Geometric representations

isotopy class a etc.

(a b)  [a b]  { a b}  (a,b)  (a,b,c)  
LS's a and b are
point-symmetry related: (a b)
translation-symmetry related: [a b}
point- and translation-symmetry related: {a b} 
LS's forming an orthogonal pair: (a b)
mutually orthogonal: (a b c) etc.

as such  (Latin square as such)
Latin square in nxn format, in matrix form 

bounded (unbounded)
Latin structure as such, in nxn format (extended on a Latin field, or put on a torus. 

colour-symmetry element  
Operation copying a coloured structure geometrically and performing a switch of two  of the colours of all corresponding symbols or a step in a colour sequence throughout the entire structure (perfect colouring)

composition, composition formula of a Latin square
specification of the types and numbers of the constituting Latin patterns of the Latin square.

conventional symmetry
geometric symmetry, disregarding black/white - colour symmetry, etc.

coordination square
Ls matrix in which the site values have been replaced by nnn (nearest neighbours number at the site)

counterpart(s), cp, (of a Latin square)
Ls or Ls’s which can be brought into geometric coincidence with a Ls under consideration by one of the operations of the dihedral group and/or/or the translation group followed by transformation towards sf. Counterparts are members of manifolds. A Ls with no counterpart is a single.

clint value
Clustering-intensity value obtained by summing nnn (nearest neighbours number) over the Ls and dividing by n^2

cyclic shifting operation on a Ls

equal (symbolically- or really different, unequal) Ls's 
Ls's A and B, with same or different nomenclature systems, are equal,the same (symbolically different) if an (no) unambiguous correspondence can be established between all corresponding symbols of A and B.The correspondence in case of equal nomenclature is a permutation. Equality under different and same nomenclature) has been illustrated in par.1.1. 
If we rewrite B in the nomenclature of A with one row or column following A, then A and B are made identical.  
If, with one corresponding row or column equal, A and B are still symbolically different then we must see whether they or their corresponding Latin fields are symmetrically related (Chapters 1 and 2). Can (or can't) they coincide, be made identical, by a combination of one of the symmetry operations of the dihedral group of order 8 (rotation, reflection etc) and/or by one of the group of n^2 translations defined on the sites lattice (shifting), including the do nothing operations of both groups? If so (if not) so, then we term A and B symmetrically related or equal, dependent, isomorphous (independent, unrelated, really different).
Terms and Notions example: in the full (single-nomenclature) 5x5 domain we have 161280 symbolically different Ls's and 18 really different Latin structures.

extended Ls's
Ls's as such, with one or more extra rows and columns added cyclically, usually for easier observation of symmetries or Latin patterns.

family, subfamily; single/double structured
Complete collection of Ls's – of the same order – sharing the same composition formula.
Families of orders up to 5 are either single- structured: all carrying the same structure type, or double-structured, with an alfa- and a beta structure type. The family can then be partitioned in an alfa- and a beta (sub)family.

Geometric Representation (GR) of a LS set of order n
preferred collection of independent Ls's selected from all manifolds within an order n set 

glide line
conventional symmetry operation: translation along the line followed halfway by reflection

identical (Ls's) 
exactly the same, coinciding

Ls, Latin square
general term, usually for Latin square in matrix form (nxn bounded), not numbered.

Latin field
Unbounded repetition of a Latin square, or tororoidal representation of a Ls

Latin-periodic structure 
2D-periodic extension of a Latin square on the (extended) sites-lattice (par.1.2)
Latin structure
Collective term for both bounded and unbounded LS representations including any ordering of site values on a bounded or extended square- or trigonal sites lattice in accordance with the Latin square condition

Latin-periodic structure 2D-periodic extension of a Latin square on the (extended) sites-lattice (par.1.2) 
LS01,...LS24;  LS(n)0..1,...LS(n)... 
Latin squares of order 4, numbered; eventually extended towards general n.

Latin pattern
Line system of connections between Ls-sites occupied by a specific symbol. Latin patterns may be extended periodically similar to Ls’s. Drawn connections may be limited for clarity to nearest - or nearest and next nearest neighbours in diagrams

Latin patterns composition diagram (briefly: composition diagram) 
Diagram showing a Ls drawn by superposition of all n Latin patterns. Composition diagrams may be extended as ongoing structures similar to Latin squares 

main (back)
direction along or parallel with the main (back) diagonal of a matrix

Ls combined with its counterparts

LS belonging to a same-order set (set-member)
also, similarly: family-member, manifold-member, GR-member

total number of point group- and/or translation-related counterparts
also: number of members in a family

neighbours / nearest neighbours / next nearest neighbours
close identical values / identical values at positions (x,y) (x+1, y+1) etc. / (x,y) (x+1) (y+2), etc. (the knight ‘s move in chess)

number of nearest neighbours of a site

preferred Ls structures 
Ls structures which have been selected from the manifolds to represent the structure types in the overall Geometric Representation (GR) of that set. Choice is arbitrary but manifold members are preferred which show symmetry and structural pattern of the Ls best 

representation representing a full same-order set if we count all point symmetry related squares as one (par. 3.4, App.III),

pointgroup related LS's:
Ls's coinciding under the action of an element of the symmetry group of a square (the dihedral group of order 8) and a  sf-permutation 

full collection of Ls’s of same order

site in a Ls without same-value nearest neighbours
also: LS without pointgroup and/or translation symmetry related counterparts

SE sequential enumeration
see Appendix II

sf ; snf;  sf-permutation
standard format(par.1.1); standard numerical format with first row in sequential order:1 2 3 ..n; operation (permutation) bringing a LS in standard format
structure type
specific structure of any member of a Geometric Representation 
Representing a full set if we count all  translation-related members as one (par. 3.4, App.III).

unequal Ls's see equal