A MINIMAL CONGRUENCE LATTICE REPRESENTATION FOR

Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.

A Residuated Lattice of L-Fuzzy Subalgebras of a Mono-Unary Algebra

Given a complete residuated lattice [Formula: see text] and a mono-unary algebra [Formula: see text], it is well known that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subsets of [Formula: see text] satisfy the same residuated lattice identities. In this paper, we show that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subalgebras of [Formula: see text] satisfy the same residuated lattice identities if and only if the Heyting algebra [Formula: see text] of subuniverses of [Formula: see text] is a Boolean algebra. We also show that [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) if and only if [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) and [Formula: see text] is a Boolean algebra.

Existence of finite bases for quasi-equations of unary algebras with 0

A finite unary algebra of finite type with a constant function 0 that is a one-element subalgebra, and whose operations have range {0, 1}, is called a {0, 1}-valued unary algebra with 0. Such an algebra has a finite basis for its quasi-equations if and only if the relation defined by the rows of the nontrivial functions in the clone form an order ideal.

Subalgebra lattices of a partial unary algebra

Necessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong subalgebra lattices of one partial unary algebra, respectively. Next, we will describe the initial segment lattice of a partial unary algebra. Applying this result, pairs of lattices of strong subalgebras and initial segments will be characterized. Further, we will characterize pairs of lattices of relative and strong subalgebras and also other pairs of subalgebra lattices of one partial unary algebra.

PRIMITIVE POSITIVE FORMULAS PREVENTING A FINITE BASIS OF QUASI-EQUATIONS

A finite unary algebra with a primitive positive formula that defines the graph of a finite group operation does not have a finite basis for its quasi-equations.

Maximal compatible extensions of partial orders

We give a complete description of maximal compatible partial orders on the mono-unary algebra (A, f), where f: A → A is an arbitrary unary operation.

HOW FINITE IS A THREE-ELEMENT UNARY ALGEBRA?

We show that, within the class of three-element unary algebras, there is a tight connection between a finitely based quasi-equational theory, finite rank, enough algebraic operations (from natural duality theory) and a special injectivity condition.

Strong and full dualisability: three-element unary algebras

We characterise the strongly dualisable three-element unary algebras and show that every fully dualisable three-element unary algebra is strongly dualisable. It follows from the characterisation that, for dualisable three-element unary algebras, strong dualisability is equivalent to a weak form of injectivity.

Mono-unary algebras are strongly dualizable

We show that mono-unary algebras have rank at most two and are thus strongly dualizable. We provide an example of a strong duality for a mono-unary algebra using an alter ego with (partial) operations of arity at most two. This mono-unary algebra has rank two and generates the same quasivariety as an injective, hence rank one, mono-unary algebra.