We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s represe...

The equality on the inverval I is the equality in the free bounded distributive lattice on generators i, 1− i. The equality in the face lattice F is the one for the free distributive lattice on formal generators (i = 0), (i = 1) with the relation (i = 0) ∧ (i = 1) = 0. We have [(r ∨ s) = 1] = (r = 1) ∨ (s = 1) and [(r∧s) = 1] = (r = 1)∧ (s = 1). An irreducible element of this lattice is a face,...

This paper shows basic properties of covariety lattices. Such lattices are shown to be infinitely distributive. The covariety lattice LCV(K) of subcovarieties of a covariety K of F -coalgebras, where F : Set → Set preserves arbitrary intersections is isomorphic to the lattice of subcoalgebras of a Pκ-coalgebra for some cardinal κ. A full description of the covariety lattice of Id-coalgebras is ...

In lattice theory, lattice polynomials have been defined as well-formed expressions involving variables linked by the lattice operations ∧ and ∨ in an arbitrary combination of parentheses. In turn, such expressions naturally define lattice polynomial functions. For instance, p(x1, x2, x3) = (x1 ∧ x2) ∨ x3 is a 3-ary lattice polynomial function. The concept of lattice polynomial function can be ...

We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially (lattice-ordered) set. Bimonoids form an appropriate framework for the study general notion complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses monoids. The central question paper is whether how bimonoids can be embedded...

Abstmct-Multilevel constructions of the binary Golay code and the Leech lattice are described. Both constructions are based upon the prvjection of the Golay code and the Leech lattice onto the (6,3,4) hexacode over GF(4). However, unlike the previously reported constructiods, the new multilevel constructions make the three levels independent by way of using a didlerent set of coset representati...

We consider superfluous elements in a bounded lattice with 0 and 1, introduce various types of graphs associated these elements. The notions such as element graph (S(L)), join intersection (JI(L)) lattice, distributive (SI(L)) are defined. Dual atoms play an important role to find connections between the lattice-theoretic properties those corresponding graph-theoretic properties. Consequently, ...