For a class $K$ of structures and $A\in K$ let ${Con}^{*}\left(A\right)$ resp. ${Con}^K\left(A\right)$ denote the lattices of $*$-congruences resp. $K$-congruences of $A$, cf. Weaver [25]. Let ${Con}^{*}\left(K\right):=I\{{Con}^{*}\left(A\right):A\in K\}$ where $I$ is the operator of forming isomorphic copies, and ${Con}^r\left(K\right):=I\{{Con}^K\left(A\right):A\in K\}$. For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${Con}^{<}\left(A\right)$, and let ${Con}^{<}\left(K\right):=I\{{Con}^{<}\left(A\right):A\in K\}$ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^s$ and $P$, respectively. Let $\lambda$ be a lattice identity and let $\Sigma$ be a set of lattice identities. Let $\Sigma {\vDash }_c\lambda (r;Q^s,P)$ denote...

This note deals with two logical topics and concerns Boolean Algebras from an elementary point of view. First we consider the class of operations on a Boolean Algebra that can be used for modelling If-then propositions. These operations, or Conditionals, are characterized under the hypothesis that they only obey to the Modus Ponens-Inequality, and it is shown that only six of them are boolean two-place functions. Is the Conditional Probability the Probability of a Conditional? This problem will...

In an algebraic frame $L$ the dimension, $dim\left(L\right)$, is defined, as in classical ideal theory, to be the maximum of the lengths $n$ of chains of primes $p_0<p_1<\cdots <p_n$, if such a maximum exists, and $\infty$ otherwise. A notion of “dominance” is then defined among the compact elements of $L$, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of $L$, including the frames $dL$ and $zL$ of $d$-elements and $z$-elements, respectively. The more concrete illustrations...

This paper continues the investigation into Krull-style dimensions in algebraic frames. Let $L$ be an algebraic frame. $dim\left(L\right)$ is the supremum of the lengths $k$ of sequences $p_0<p_1<\cdots <p_k$ of (proper) prime elements of $L$. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of $L$ in terms of the dimensions of certain boundary quotients of $L$. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...