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Vassilis G. Kaburlasos (Technological Educational Institution of Kavala, Greece)

DOI: 10.4018/978-1-59904-849-9.ch181

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TopLattice theory has been compiled by Birkhoff (Birkhoff, 1967). This section summarizes selected results regarding a Cartesian product lattice (L,≤)= (L_{1},≤_{1})×…×(L_{N},≤_{N}) of *constituent* lattices (L_{i},≤_{i}), i=1,…,N.

Given an *isomorphic* function θ_{i}: (L_{i},≤_{i})→(L_{i},≤_{i})^{∂} in a constituent lattice (L_{i},≤_{i}), i=1,…,N, where (L_{i},≤_{i})^{∂} ≡ (L_{i},≤) denotes the *dual* (lattice) of lattice (L_{i},≤_{i}), then an isomorphic function θ: (L,≤)→(L,≤)^{∂} is given by θ(*x*_{1},…,*x*_{N})=(θ_{1}(*x*_{1}),…,θ_{N}(*x*_{N})).

Given a *positive valuation* function *v*_{i}: (L_{i},≤_{i})→R in a constituent lattice (L_{i},≤_{i}), i=1,…,N then a positive valuation *v*: (L,≤)→R is given by *v*(*x*_{1},…,*x*_{N})=*v*_{1}(*x*_{1})+…+*v*_{N}(*x*_{N}).

It is well-known that a positive valuation *v*_{i}: (L_{i},≤_{i})→R in a lattice (L_{i},≤_{i}) implies a metric function *d*_{i}: L_{i}×L_{i}→ given by *d*_{i}(*a*,*b*) = *v*_{i}(*a*∨*b*) - *v*_{i}(*a*∧*b*).

*Minkowski metrics d _{p}*: (L

Dual (Lattice): Given a lattice (L,=), its dual lattice, symbolically (L,=)? or (L,=?) = (L,=), is a lattice with the inverse order relation (=).

Isomorhic (Function): Given two lattices (L1,=1) and (L2,=2), an isomorphic function is a bijective (one-to-one) function f: (L1,=1)?(L2,=2) such that x=y ? f(x)=f(y).

ART: ART stands for Adaptive Resonance Theory. That is a biologically inspired neural paradigm for, originally, clustering binary patterns. An analog pattern version of ART, namely fuzzy-ART, is applicable in the unit hypercube. The corresponding neural network for classification is called fuzzy-ARTMAP.

Rule Induction: Process of learning, from cases or instances, if-then rule relationships that consist of an antecedent (if-part, defining the preconditions or coverage of the rule) and a consequent (then-part, stating a classification, prediction, or other expression of a property that holds for cases defined in the antecedent).

Positive Valuation (Function): Given a lattice (L,=), a positive valuation is a function v: (L,=)?R, which satisfies both v(x)+v(y) = v(x?y)+v(x?y) and x

Subattice: A sublattice (S,=) of a lattice (L,=) is another lattice such that both S?L and x,y?S ? x?y,x?y?S.

Lattice: A lattice is a poset (L,=) any two of whose elements have both a greatest lower bound (g.l.b.), denoted by x?y, and a least upper bound (l.u.b.), denoted by x?y.

FIS: FIS stands for Fuzzy Inference System. That is an architecture for reasoning involving fuzzy sets (typically fuzzy numbers) based of fuzzy logic.

Poset: A partially ordered set (or, poset, for short) is a pair (P,=), where P is a set and = is an order relation on P. The latter (relation) by definition satisfies (1) x=x, (2) x=y and y=x ? x = y, and (3) x=y and y=z ? x=z.

SOM: SOM stands for Self-Organizing Map. That is a biologically inspired neural paradigm for clustering analog patterns. SOM is often used for visualization of nonlinear relations of multi-dimensional data.

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