This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with...

This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with...

The paper contains some sufficient conditions for Marczewski-Burstin representability of an algebra 𝓐 of sets which is isomorphic to 𝓟(X) for some X. We characterize those algebras of sets which are inner MB-representable and isomorphic to a power set. We consider connections between inner MB-representability and hull property of an algebra isomorphic to 𝓟 (X) and completeness of an associated quotient algebra. An example of an infinite universally MB-representable algebra is given.

We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $\langle a_{\xi }:\xi <\alpha \rangle$ of elements of $A$, with $\alpha$ an ordinal, such that for all $F,G\in {\left[\alpha \right]}^{<\omega }$ with $F<G$ we have ${\prod }_{\xi \in F}{a}_{\xi }·{\prod }_{\xi \in G}-a_{\xi }\ne 0$. A free sequence of length $\alpha$ exists iff the Stone space $Ult\left(A\right)$ has a free sequence of length $\alpha$ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $${𝔣}_{sp}\left(A\right)=\left\{\right|\alpha |:A\text{has}\text{an}\text{infinite}\text{maximal}\text{free}\text{sequence}\text{of}\text{length}\alpha \}$$ and the associated min-max function $$𝔣\left(A\right)=min\left({𝔣}_{sp}\left(A\right)\right).$$ Among the results...

Let τ be a type of algebras without nullary fundamental operation symbols. We call an identity φ ≈ ψ of type τ clone compatible if φ and ψ are the same variable or the sets of fundamental operation symbols in φ and ψ are nonempty and identical. For a variety of type τ we denote by ${}^c$ the variety of type τ defined by all clone compatible identities from Id(). We call ${}^c$ the clone extension of . In this paper we describe algebras and minimal generics of all subvarieties of ${}^c$, where is the variety of...

In a Boolean Algebra B, an inequality f(x,x --&gt; y)) ≤ y satisfying the condition f(1,1)=1, is considered for defining operations a --&gt; b among the elements of B. These operations are called Conditionals'' for f. In this paper, we obtain all the boolean Conditionals and Internal Conditionals, and some of their properties as, for example, monotonicity are briefly discussed.

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for ${\aleph }_{\omega }$, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals,...

Semirings are modifications of unitary rings where the additive reduct does not form a group in general, but only a monoid. We characterize multiplicatively idempotent semirings and Boolean rings as semirings satisfying particular identities. Further, we work with varieties of enriched semirings. We show that the variety of enriched multiplicatively idempotent semirings differs from the join of the variety of enriched unitary Boolean rings and the variety of enriched bounded distributive lattices....