Finite Unary Algebra Research Articles

Coalgebraic methods for Ramsey degrees of unary algebras

Abstract In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, and using pre-adjunctions transport the Ramsey properties, we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

On the number of countable subdirect powers of unary algebras

A finite unary algebra [Formula: see text] has only countably many countable subdirect powers if and only if every operation [Formula: see text] is either a permutation or a constant mapping.

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.

All Congruence Modular Symmetric and Near-Symmetric Algebras

For a unary operation f on a finite set A , let denote λ ( f ) the least non-negative integer withIm f λ ( f ) = Im f λ ( f )+1which is called the pre-period of f . K. Denecke and S. L. Wismath have characterizedall operations f on A with λ ( f ) = A −1 and prove that λ ( f ) = A −1 if and only if there exists a d ∈ Asuch that A = {d, f (d), f 2 (d), , f A 1(d)} where f A 1(d) f A (d) − = . C. Ratanaprasert and K. Deneckehave characterized all operations f on A with λ ( f ) = | A | −2 for all | A | ≥ 3; and have characterizedall equivalence relations on A which are invariant under f with these long pre-periods.In the paper, we study finite unary algebras A = (A; f ) with λ ( f )∈{0, 1} for | A | ≥ 3which are called symmetric algebras and near-symmetric algebras, respectively. We characterizeall operations f whose A is congruence modular. We prove that a symmetric algebra A iscongruence modular if and only if the lattice ConA of all congruence relations is either a product ofchains or a linear sum of a product of chains with one element top or a 3 M − head lattice; and anear-symmetric algebra A is congruence modular if and only if ConA is one of the followings:2× P, 2×(P⊕1), 2× L, 3 M × P, 3 M ×(P⊕1), or 3 M × Lwhere P denote a product of chains and L is a 3 M − head lattice. Key Words: Monounary algebra; Congruence distributive; Congruence modular

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.

Homomorphic images of finite subdirectly irreducible unary algebras

We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty.

Globals of unary algebras

In the present paper, we investigate the question on the reconstructability (up to isomorphism) of a finite unary algebra from its global. It is proved that the class of all finite unary algebras is not globally determined.

Positive primitive formulas preventing enough algebraic operations

A finite unary algebra with a positive primitive formula defining a pp-acyclic relation does not have enough algebraic operations and does not have a finite basis for its quasi-equations.

Inherent dualisability

Beginning from any finite unary algebra with at least two fundamental operations, there is an infinite ascending chain of finite algebras that are alternately dualisable and non-dualisable. We obtain this result while characterising the finite algebras that can be embedded into a non-dualisable algebra. As an aside, we give an example of a non-dualisable algebra that can be constructed by adding a constant operation to a dualisable algebra.

Binary homomorphisms and natural dualities

We investigate ways in which certain binary homomorphisms of a finite algebra can guarantee its dualisability. Of particular interest are those binary homomorphisms which are lattice, flat-semilattice or group operations. We prove that a finite algebra which has a pair of lattice operations amongst its binary homomorphisms is dualisable. As an application of this result, we find that every finite unary algebra can be embedded into a dualisable algebra. We develop some general tools which we use to prove the dualisability of a large number of unary algebras. For example, we show that the endomorphisms of a finite cyclic group are the operations of a dualisable unary algebra.

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pio1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A.

Cancellation among finite unary algebras

We show that a unary algebra is cancellable among finite unary algebras if and only if it contains a one-element subalgebra.

Quasiidentities of finite unary algebras

Quasiidentities of finite unary algebras

Families of independent sets in finite unary algebras

Families of independent sets in finite unary algebras

Subdirectly Irreducible Unary Algebras

Abstract : Subdirect representations of finite unary algebras are studied. Elementary considerations of the corresponding directed graphs yield necessary and sufficient conditions for connected finite unary algebras to be irreducible, i.e., to have only trivial subdirect representations. (Author)