On Lattice Implication Algebra And Its Product Structure

Lattice implication algebra is one of the truth-valued field of lattice-valued propositional logic and first-order logic. Based on existing researches, this paper will further discuss the duality, dual atom, filter generated by a Boolean element and lattice implication product algebra. In the meantime, certain conclusions on lattice implication algebra are perfected. The main results of this paper are as follows:1. Principle of duality in lattice implication algebra is given. Further, details of dual processes are shown by some dual concepts and theorems.2. Certain conclusions in [37] and [39] are perfected and the new ones are as follows.(1) In [39], the conclusion "if a is the unique dual atom of lattice implication algebra L with ord(a)=p (finite) and A={an|n=0,1,...,p}, then bâ†'x=bâ†'y ((?)x, b∈A, y∈L) implies x=y" can’t be got.(2) If a is the unique dual atom of a finite lattice implication algebra L and ord(a)=p1,...,p}hold.(3) If a is the unique dual atom of a finite lattice implication algebra L and ord(a)=p (finite), then L (?) L(p).(4) Point out that in the proof for Expression4.1.4, should be revised into3. Some properties of dual atom are obtained. It proves that in a finite lattice implication algebra L, L is a chain if and only if tnt=O holds for any dual atom t of L with ord(t)=n1(finite). In addition, it concludes that in lattice implication product algebra L1×L2, Dat(L1×L2)={(a1, a2)|a1=I1, a2∈Dat(L2); a1∈Dat(L1), a2=I2}. Here, Dat(L) means the set of dual atoms of lattice implication algebra L.4. Two equivalent forms of filter generated by a Boolean element are shown. That means if x is a Boolean element of lattice implication algebra L=(L,∨,∧,’,â†', O, I), then [x)={y|y≥x,y∈L}={w|aâ†'x=w, a∈L}. In addition, relationship between Boolean elements of L1×L2and those of L1and L2is given, which is B(L1×L2)={(x1, x2)|x1∈B(L1),x2∈B(L2)}. Here, B(L) means the set of Boolean elements of lattice implication algebra L. 5. Concrete kinds of filters (including implicative filters, proper filters, prime filters, ultra-filters, obstinate filters, I-filters, involution filters, extended filters and weak filters) of lattice implication product algebra are discussed. Concretely, the relationships between the above-mentioned filters of L1×L2and those of L1and L2are studied. Then the study is extended to L1×L2×...×Ln and specialized into Lm×Ln. After that, the applications of the above theories are shown.(1) The structures and some kinds of filters of L6, L8and L9are shown;(2) Positions of non-trivial filters are shown respectively in the Hasse diagrams of two-dimensional and three-dimensional finite lattice implication algebras (namely, Lm×Ln and Lm×Ln×Lk);(3) It is obtained that Lm (1) is a lattice H implication algebra.