An effect algebraic partial binary operation $øplus$ defined on the underlying set $E$ uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion $\widehat{E}$ of $E$ there exists an effect algebraic partial binary operation $\widehat{\oplus }$ then $\widehat{\oplus }$ need not be an extension of $\oplus$. Moreover, for an Archimedean atomic lattice effect algebra $E$ we give a necessary and sufficient condition for that $\widehat{\oplus }$ existing on $\widehat{E}$ is an extension of $\oplus$ defined on $E$. Further we show that such $\widehat{\oplus }$ extending $\oplus$ exists at most...

Join-independent and meet-independent sets in complete lattices were defined in [6]. According to [6], to each complete lattice (L,≤) and a cardinal number p one can assign (in a unique way) an incidence structure $J_L^p$ of independent sets of (L,≤). In this paper some lattice-inadmissible incidence structures are founded, i.e. such incidence structures that are not isomorphic to any incidence structure $J_L^p$.

A dcpo $P$ is continuous if and only if the lattice $C\left(P\right)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C\left(P\right)$. In this paper, we study the order-theoretic properties of $C\left(P\right)$ for general dcpo’s $P$. The main results are: (i) every $C\left(P\right)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C\left(P\right)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices...

Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we firstly introduce the concepts of left and right semi-uninorms on a complete lattice and illustrate these notions by means of some examples. Then, we lay bare the formulas for calculating the upper and lower approximation left (right) semi-uninorms of a binary...