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1

Retracts of ultrahomogeneous structures in the context of Katětov functors

Mašulović D.

Demonstratio Mathematica

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2015

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Vol. 48, nr 4

562--577

EN

In this paper, we characterize retracts of a wide class of Fraïssé limits using the tools developed in a recent paper by W. Kubiś and the present author, which we refer to as Katětov functors. This approach enables us to conclude that in many cases, a structure is a retract of a Fraïssé limit if and only if it is algebraically closed in the surrounding category.

2

Monounary algebras with same quasiorders or retracts

Jakubíková-Studenovská D., Petrejčíková M., Pócs J.

Demonstratio Mathematica

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2011

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Vol. 44, nr 3

481-496

EN

Let (A, f) be a monounary algebra. We describe all monounary algebras (A, g) having the same set of quasiorders, Quord (A, f) = Quord (A, g). It is proved that if Quord (A, f) does not coincide with the set of all reflexive and transitive relations on the set A and (A, f) contains no cycle with more than two elements, then f is uniquely determined by means of Quord (A, f). In the opposite case, Quord (A, f) = Quord (A, g) if and only if Con (A, f) = Con (A, g). Further, we show that, except the case when Quord (A, f) coincides with the set of all reflexive and transitive relations, if the monounary algebras (A, f) and (A, g) have the same quasiorders, then they have the same retracts. Next we characterize monounary algebras which are determined by their sets of retracts and connected monounary algebras which are determined by their sets of quasiorders.

3

A Proof of Plotkin's Conjecture

Yinbin L., Maokang L.

Fundamenta Informaticae

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2009

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Vol. 92, nr 3

301-306

EN

In 1978, G. Plotkin [7] conjectured that for the three-element truthvalue dcpo T, if k> ω then the function space [T^k → T^k] is not a retract of T^k. In this short paper, we constructively prove a stronger result that if k>ω then the function space [T^k → T^k] is not a retract of the Cartesian product of any family of finite posets. Thus Plotkin's Conjecture is proved to be correct.

4

On a complete lattice of retracts of a free monoid generated by three elements

Foryś W.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

123-127

EN

We prove that the family of retracts of a free monoid generated by three elements, partially ordered with respect to the inclusion, is a complete lattice.

5

Retracts of Algebras

Novotny M.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

375-384

EN

Construction of retracts of a general algebra is described by using retracts of mono-unary algebras. Inspirational influence of the theory of Pawlak machines is mentioned.

6

A retract principle on discrete time scales

Diblik J., Ruzickova M., Vaclavikova B.

Opuscula Mathematica

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2006

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Vol. 26, no. 3

445-455

EN

In this paper we discuss asymptotic behavior of solutions of a class of scalar discrete equations on discrete real time scales. A powerful tool for the investigation of various qualitative problems in the theory of ordinary differential equations as well as delayed differential equations is the retraction method. The development of this method is discussed in the case of the equation mentioned above. Conditions for the existence of a solution with its graph remaining in a predefined set are formulated. Examples are given to illustrate the results obtained.

7

A note on absolute approximate retracts

Martin J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 4

355-359

EN

The notion of an absolute approximate retract for a class Q of topological spaces (or an AAR(Q)-space) generalizes the concept of an absolute retract for the class Q. For many classes Q, it is shown that AAR(Q)-spaces are preserved under retraction mappings and that a fully normal AAR(Q)-space X must be contractible and can be expressed as a product of finite-dimensional compacta if and only if X is homeomorphic to a cube or is a finite-dimensional AR-space.