**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 pairas such(Latin square as such) Latin square in nxn format colour-symmetry elementoperation copying a LS structure geometrically and performing a switch of two or more of the site symbols throughout the entire LScomposition, composition formula (of a Latin square)specification of the types and numbers of the constituting Latin patterns of the Latin square.conventional symmetrysymmetry disregarding colour symmetrycoordination squareLS matrix in which the value of each site values has been replaced by nnn (nearest neighbours number)counterpart (of a Latin square)LS or LS’s which can be brought into geometric coincidence with a LS under consideration by one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 thedihedral groupand/or/or the translation group followed by the sf transformation. Counterparts can be collected in manifolds. A LS with no counterpart is asingle.Clint valueClustering intensity value obtained by summingnnn(nearest neighbours number) over the square and dividing by n^2cshcyclic shifting operationdependent, seesymmetry relatedequal, ,(different) LS's, symbolicaly differentLS'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 differentLS'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 couplesextended LS'sLS'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 samecomposition 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 preferredcollection of independent LS'sselectedfrom allmanifoldswithin an order n setglide linesymmetry operation: translation along the line followed halfway by reflectionpreferred LS'sHigh(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 setidentical(LS's) exactly the same, coincidingLS, Latin square, Latin square as suchnxn bounded Latin squareLatin fieldUnbounded repetition of a Latin square, ortororoidal representationof a LSLatin structureCollective term for both bounded and unbounded LS representationsLSi (1…i….24)Latin square of order 4, numbered (Lsn) / (nsf) various numbered representations of Latin squares / under symmetry operation (sfetc) (par. 1.1 and 1.4 only)Latin patternLinesystem 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 diagramsLatin 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 squaresLatin-periodic structure,brieflyLatin field2D - periodic extension of a Latin square on the (extended)sites-lattice(par. 1.2)manifoldLS combined with itscounterparts memberLS belonging to a same-order set (set-member) also, similarly: family-member, manifold-member, GR-membermultiplicitytotal number of point group- and/or translation-related counterparts also: number of members in a same composition familynearest neighbours / next nearest neighboursidentical symbols at positions (x, y) (x+1, y+1) etc. / (x, y) (x+1) (y+2), etc. (the knight ‘s move in chess)nnnnumber of nearest neighbours of a sitePR-representationrepresentation 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 permutationset: full collection of LS’s of same ordersymmetryrelated, 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 singlesite in a LS without same-value nearest neighbours also: LS without point group and/or translation symmetry related counterpartsSE sequential enumerationsee Appendix IIsf ; snf; sf-permutationstandard format(par.1.1); first row in sequential order:1 2 3 ….. operation (permutation) bringing a LS in standard formatstructureordering of site values in a nxn or extended lattice in accordance with the Latin square condition.structure typespecific structure of any member of aGeometric RepresentationTR-representationRepresenting a full set if we count all l translation-related members as one (par. 2.3).