**Appendix I . Terms and notions**

1....5, 1...18numbering of order 4 and 5 Ls's which we preferred for representing thestructure typesin the correspondingGeometric representations (a)isotopy class a etc.(a b) [a b] { a b} (a,b) (a,b,c) LS's a and b arepoint-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 formbounded (unbounded) Latin structure as such, in nxn format (extended on a Latin field, or put on a torus.colour-symmetry elementOperation 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 squarespecification of the types and numbers of the constitutingLatin patternsof the Latin square.conventional symmetrygeometric symmetry, disregarding black/white - colour symmetry, etc.coordination squareLs matrix in which the site values have been replaced bynnn(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 thedihedral groupand/or/or the translation group followed by transformation towardssf.Counterparts are members ofmanifolds.A Ls with no counterpart is asingle.clint valueClustering-intensity value obtained by summingnnn(nearest neighbours number) over the Ls and dividing by n^2cshcyclic shifting operation on a Lsequal (symbolically- or really different, unequal) Ls'sLs'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 aremadeidentical. 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 differentLatin structures.extended Ls'sLs's as such, with one or more extra rows and columns addedcyclically, 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 samecomposition formula. Families of orders up to 5 are eithersingle- structured: all carrying the same structure type, ordouble-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 preferredcollection ofindependent Ls's selectedfrom allmanifoldswithin an order n setglide line conventionalsymmetry operation: translation along the line followed halfway by reflectionidentical(Ls's) exactly the same, coincidingLs, Latin squaregeneral term, usually for Latin square in matrix form (nxn bounded), not numbered.Latin fieldUnbounded repetition of a Latin square, ortororoidal representationof a LsLatin-periodic structure2D-periodic extension of a Latin square on the (extended) sites-lattice (par.1.2)Latin structureCollective term for both bounded and unbounded LS representations including any ordering of site values on a bounded or extended square- ortrigonal sites latticein accordance with the Latin square conditionLatin-periodic structure2D-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 Linesystem 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 clarityto nearest - or nearest and next nearest neighboursin diagramsLatin 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 squaresmain (back)direction along or parallel with the main (back) diagonal of a matrixmanifoldLs combined with itscounterparts memberLS belonging to a same-order set (set-member) also, similarly: family-member, manifold-member, GR-membermultiplicitytotal number ofpoint group- and/or translation-related counterpartsalso: number of members in a familyneighbours /nearest neighbours / next nearest neighboursclose 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)nnnnumber ofnearest neighboursof a sitepreferred Ls structuresLs structures which have been selected from themanifoldsto represent thestructure typesin the overallGeometric Representation(GR) of that set. Choice is arbitrary but manifold members are preferred which show symmetry and structural pattern of the Ls bestPR-representationrepresentation 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-permutationset: full collection of Ls’s of same ordersinglesite in a Ls without same-valuenearest neighboursalso: LS without pointgroup and/or translation symmetry related counterpartsSE sequential enumerationsee Appendix IIsf ; snf; sf-permutationstandard format(par.1.1); standard numerical format with first row in sequential order:1 2 3 ..n; operation (permutation) bringing a LS in standard formatstructure typespecific structure of any member of aGeometric RepresentationTR-representationRepresenting a full set if we count all translation-related members as one (par. 3.4, App.III).unequal Ls'sseeequal