**Terms and notions**

*(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) form an orthogonal pair

*(1) etc (1sf), (1csh) etc.; (a)*

(1) etc. ; a specific LS ; (1sf) specific LS under or after a (symmetry movement

(a) is numbered isotopy class .

*basic structure
*A structure which has been adopted as a member of a geometric representation of a LS-set of fixed order.

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

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

*conventional symmetry
*symmetry disregarding colour symmetry

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*LS matrix in which the site values have been replaced by their nnn (nearest neighbours number) at the site

*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. Counterparts can be collected in manifolds. A LS with no counterpart is a single.

*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

*Clint value
*Clustering intensity value obtainned by summing nnn over the square and dividing by n^2

*csh
*cyclic shifting operation

*dependent*, see *symmetry related*

*equal, ,(different) LS’s
*LS’s A and B with different nomenclature systems, are equal, (different) if a (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.

Note that different LS’s can be different cuttings from the same Latin periodic structure. In the 5×5 domain we have 1344 different LS’s and 18 different (Latin-periodic) structures.

*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 to coincide by a combination of one of the symmetry operations of the dihedral group of order 8 and one of the group of n^2 translations defined on the sites lattice, including the “do nothing” .operations of both groups

*extending a LS : *adding one or more rows and/or columns to the LS’s, or entire copies of the LS’s, usually for better observation of symmetries 0r Latin patterns

*family, subfamily
*Collection of all 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
*If any LS of order n is symmetry related with exactly one member of a set of Latin squares then that set is a geometric representation of the entire order n set

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

*LS, Latin square, Latin square as such*

nxn bounded Latin square

*Latin structure
*Unbounded repetition of a Latin square, or tororoidal representation of a LS

*LS structure
*Collective term for both bounded and unbounded LS representations

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

*Latin pattern
Line s*ystem 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-periodic structure (of order n)
*2D-periodic occupation of all sites of a lattice with periods n by symbols of n different kinds conforming with the Latin square condition.. All isometric transformations products of that distribution are considered identical.

Simpler: 2D- periodic extension of a Latin square on the (extended) sites-lattice (par. 1.2)

*Latin structure
*Shorthand for Latin-periodic structure; also representation of a LS on the torus

*manyfold
*LS combined with its counterparts

*member
*LS belonging to a same-order set

also: belonging to a family of LS’s (family-member)

also: belonging to a Geometric Representation (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

*
same* (LS, LS-structure)

see different

*set*: full collection of LS’s of same order

*single
*site in a LS without same-value nearest neighbours

also: LS without point group – or translation symmetry related counterparts

*SE sequential enumeration
*see Appendix II

*sf* standard format

*snf, standard numeric format of a LS
*LS with first row 1 2 3 …..

*structure
*ordering of site values in a nxn or extended lattice in accordance with the Latin square condition. The structure can be adopted by a LS as such or by a Latin periodic structure

*structure type
*Latin structure of any member of a Geometric Representation , or a numbered LS isomorphous with a numbered member of a Geometric Representation,

*TR-representation
*Representing the full set if we count all l translation-related members as one (par. 2.3).