Appendix I

Appendix I . Terms and notions


(a) 
isotopy class a etc

(a b)  [a b]  { a b}  (a,b)    LS's a and b are
 (a b) point symmetry related ; [a b] translation symmetry related; {a b} point and translation symmetry related; (a,b) LS's forming an orthogonal pair

 as such  (Latin square as such)                                                                      Latin square in nxn format   

colour-symmetry element
operation copying a LS structure geometrically and performing a switch of two or more of the site symbols throughout the entire LS

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

conventional symmetry
symmetry disregarding colour symmetry

coordination square
 LS matrix in which the  value of each site values has been replaced by nnn (nearest neighbours number)

counterpart (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 the sf transformation. Counterparts can be collected in manifolds. A LS with no counterpart is a single.

Clint value
Clustering intensity value obtained by summing nnn (nearest neighbours number) over the square and dividing by n^2

csh
 cyclic shifting operation

dependent, see symmetry related

equal, ,(different) LS's, symbolicaly different
 LS's A and B with different nomenclature systems, are equal, (different) if an (no) unambiguous correspondence can be established between these systems. If the nomenclature systems is the same then the unambiguous correspondence is a permutation. Both cases (different and same nomenclature) have been illustrated in par.1.1.
symbolically different LS's must be put on standard format to see whether they are identical, equal or unrelated.In the order 4 domain we have 576 numerically different...different couples and ....symmetry related couples

extended LS's
LS's with  one or more rows and/or columns added,usually for better 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 a 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
 symmetry operation: translation along the line followed halfway by reflection

preferred LS's 
High(est) symmetry LS's which have been selected from the manifolds of a fixed order LS-set to form a Geometric Representation (GR) of that set

identical (LS's) 
exactly the same, coinciding

LS, Latin square, Latin square as such
nxn bounded Latin square

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

Latin structure
Collective term for both bounded and unbounded LS representations

LSi (1…i….24)
Latin square of order 4, numbered

(Lsn) / (nsf)
various numbered representations of Latin squares / under symmetry operation (sf etc)
 (par. 1.1 and 1.4 only)
Latin pattern
Line system of connections in a LS of sites occupied by a specific symbol. Latin patterns may be extended two-dimensionally 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 of that LS.. Composition diagrams may be extended as ongoing structures similar to Latin squares 

Latin-periodic structure, briefly Latin field 
2D - periodic extension of a Latin square on the (extended) sites-lattice (par. 1.2)

manifold
 LS combined with its counterparts

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

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

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

nnn
 number of nearest neighbours of a site

PR-representation
 representation representing the full set if we count all point symmetry related squares as one (par. 2.2),

point group 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 permutation 

set:  full collection of LS’s of same order

symmetry related, dependent (unrelated, independent) structures of LS's
 Structures of LS's and Latin structures are symmetry-related, dependent,( symmetry-unrelated, independent) if the LS's or the corresponding Latin -periodic structures can (cannot) be made identical by a combination of 
-- one of the symmetry operations of the dihedral group of order 8
-- one of the group of n^2 translations defined on the sites lattice
-- including the “do nothing” .operations of both groups 
-- and a sf -permutation

single
 site in a LS without same-value nearest neighbours
 also: LS without point group and/or translation symmetry related counterparts

SE sequential enumeration
 see Appendix II

sf ; snf;  sf-permutation
standard format(par.1.1);
first row in sequential order:1 2 3 …..
operation (permutation) bringing a LS in standard format

structure
ordering of site values in a nxn or extended lattice in accordance with the Latin square condition.
 
structure type
specific structure of any member of a Geometric Representation  
TR-representation
 Representing a full set if we count all l translation-related members as one (par. 2.3).