Residuated operators in complemented posets

Using the operators of taking upper and lower cones in a poset with a unary operation, we define operators [Formula: see text] and [Formula: see text] in the sense of multiplication and residuation, respectively, and we show that by using these operators, a general modification of residuation can be introduced. A relatively pseudocomplemented poset can be considered as a prototype of such an operator residuated poset. As main results, we prove that every Boolean poset as well as every pseudo-orthomodular poset can be organized into a (left) operator residuated structure. Some results on pseudo-orthomodular posets are presented which show the analogy to orthomodular lattices and orthomodular posets.

The zero divisor graphs of Boolean posets

In this paper, it is proved that if B is a Boolean poset and S is a bounded pseudocomplemented poset such that S\Z(S) = {1}, then Γ(B) ≌ Γ(S) if and only if B ≌ S. Further, we characterize the graphs which can be realized as zero divisor graphs of Boolean posets.

Prime, irreducible elements and coatoms in posets

In this paper, some properties of prime elements, pseudoprime elements, irreducible elements and coatoms in posets are investigated. We show that the four kinds of elements are equivalent to each other in finite Boolean posets. Furthermore, we demonstrate that every element of a finite Boolean poset can be represented by one kind of them. The example presented in this paper indicates that this result may not hold in every finite poset, but all the irreducible elements are proved to be contained in each order generating set. Finally, the multiplicative auxiliary relation on posets and the notion of arithmetic poset are introduced, and some properties about them are generalized to posets.

REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

Starting with the zero-square "zeon algebra," the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard–Sylvester matrices are shown as well.