ENDOMORPHISMS OF DISTRIBUTIVE LATTICES WITH A QUANTIFIER

2007 ◽  
Vol 17 (07) ◽  
pp. 1349-1376 ◽  
Author(s):  
M. E. ADAMS ◽  
W. DZIOBIAK
Keyword(s):  
Boolean Lattice ◽  
Distributive Lattices ◽  
Endomorphism Monoid ◽  
Proper Class ◽  
Endomorphism Monoids

Let V be a non-trivial variety of bounded distributive lattices with a quantifier, as introduced by Cignoli in [7]. It is shown that if V does not contain the 4-element bounded Boolean lattice with a simple quantifier, then V contains non-isomorphic algebras with isomorphic endomorphism monoids, but there are always at most two such algebras. Further, it is shown that if V contains the 4-element bounded Boolean lattice with a simple quantifier, then it is finite-to-finite universal (in the categorical sense) and, as a consequence, for any monoid M, there exists a proper class of non-isomorphic algebras in V for which the endomorphism monoid of every member is isomorphic to M.

scholarly journals Pseudocomplemented distributive lattices with small endomorphism monoids

1983 ◽  
Vol 28 (3) ◽  
pp. 305-318 ◽  
Author(s):  
M.E. Adams ◽  
V. Koubek ◽  
J. Sichler
Keyword(s):  
The Other ◽  
Distributive Lattices ◽  
Endomorphism Monoid ◽  
Proper Class ◽  
Lattice Of Varieties ◽  
Endomorphism Monoids ◽  
Other Hand ◽  
Image Position

By a result of K.B. Lee, the lattice of varieties of pseudo-complemented distributive lattices is the ω + 1 chainwhere B−1, B0, B1 are the varieties formed by all trivial, Boolean, and Stone algebras, respectively. General theorems on relative universality proved in the present paper imply that there is a proper class of non-isomorphic algebras in B3 with finite endomorphism monoids, while every infinite algebra from B2 has infinitely many endomorphisms. The variety B4 contains a proper class of non-isomorphic algebras with endomorphism monoids consisting of the identity and finitely many right zeros; on the other hand, any algebra in B3 with a finite endomorphism monoid of this type must be finite.

scholarly journals Kleene algebras are almost universal

1986 ◽  
Vol 34 (3) ◽  
pp. 343-373 ◽  
Author(s):  
M. E. Adams ◽  
H. A. Priestley
Keyword(s):  
Boolean Algebras ◽  
Endomorphism Monoid ◽  
Single Element ◽  
Proper Class ◽  
Kleene Algebra ◽  
Endomorphism Monoids ◽  
De Morgan Algebras ◽  
Universal Variety

This paper studies endomorphism monoids of Kleene algebras. The main result is that these algebras form an almost universal variety k, from which it follows that for a given monoid M there is a proper class of non-isomorphic Kleene algebras with endomorphism monoid M+ (where M+ denotes the extension of M by a single element that is a right zero in M+). Kleene algebras form a subvariety of de Morgan algebras containing Boolean algebras. Previously it has been shown the latter are uniquely determined by their endomorphisms, while the former constitute a universal variety, containing, in particular, arbitrarily large finite rigid algebras. Non-trivial algebras in K always have non-trivial endomorphisms (so that universality of K is ruled out) and unlike the situation for de Morgan algebras the size of End(L) for a finite Kleene algebra L necessarily increases as |L| does. The paper concludes with results on endomorphism monoids of algebras in subvarieties of the variety of MS-algebras.

Idempotent rank in endomorphism monoids of finite independence algebras

2007 ◽  
Vol 137 (2) ◽  
pp. 303-331 ◽  
Author(s):  
R. Gray
Keyword(s):  
Proper Ideal ◽  
Endomorphism Monoid ◽  
Endomorphism Monoids ◽  
Idempotent Rank

In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

scholarly journals A universality result for endomorphism monoids of some ultrahomogeneous structures

2012 ◽  
Vol 55 (3) ◽  
pp. 635-656 ◽  
Author(s):  
Igor Dolinka ◽  
Dragan Mašulović
Keyword(s):  
Endomorphism Monoid ◽  
Sufficient Condition ◽  
Urysohn Space ◽  
Endomorphism Monoids ◽  
General Sufficient Condition ◽  
Countably Infinite ◽  
New Applications

AbstractWe devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fraïssé limit) embeds all countable semigroups. This approach not only provides us with a framework unifying the previous scattered results in this vein, but actually yields new applications for endomorphism monoids of the (rational) Urysohn space and the countable universal ultrahomogeneous semilattice.

Unicyclic graphs with regular endomorphism monoids

2016 ◽  
Vol 08 (02) ◽  
pp. 1650020 ◽  
Author(s):  
Xiaobin Ma ◽  
Dein Wong ◽  
Jinming Zhou
Keyword(s):  
Unicyclic Graph ◽  
Endomorphism Monoid ◽  
Unicyclic Graphs ◽  
Endomorphism Monoids ◽  
Odd Cycle ◽  
Open Question

The motivation of this paper comes from an open question: which graphs have regular endomorphism monoids? In this paper, we give a definitely answer for unicyclic graphs, proving that a unicyclic graph [Formula: see text] is End-regular if and only if, either [Formula: see text] is an even cycle with 4, 6 or 8 vertices, or [Formula: see text] contains an odd cycle [Formula: see text] such that the distance of any vertex to [Formula: see text] is at most 1, i.e., [Formula: see text]. The join of two unicyclic graphs with a regular endomorphism monoid is explicitly described.

scholarly journals Sunflowers in Lattices

10.37236/1963 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Geoffrey McKenna
Keyword(s):  
Finite Field ◽  
Boolean Lattice ◽  
Distributive Lattices ◽  
Sunflower Lemma

A Sunflower is a subset $S$ of a lattice, with the property that the meet of any two elements in $S$ coincides with the meet of all of $S$. The Sunflower Lemma of Erdös and Rado asserts that a set of size at least $1 + k!(t-1)^k$ of elements of rank $k$ in a Boolean Lattice contains a sunflower of size $t$. We develop counterparts of the Sunflower Lemma for distributive lattices, graphic matroids, and matroids representable over a fixed finite field. We also show that there is no counterpart for arbitrary matroids.

scholarly journals The Bergman property for endomorphism monoids of some Fraïssé limits

2014 ◽  
Vol 26 (2) ◽  
Author(s):  
Igor Dolinka
Keyword(s):  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Permutation Groups ◽  
Endomorphism Monoid ◽  
Order Structure ◽  
First Order ◽  
Endomorphism Monoids ◽  
Countably Infinite

AbstractBased on an idea of Y. Péresse and some results of Maltcev, Mitchell and Ruškuc, we present sufficient conditions under which the endomorphism monoid of a countably infinite ultrahomogeneous first-order structure has the Bergman property. This property has played a prominent role both in the theory of infinite permutation groups and, more recently, in semigroup theory. As a byproduct of our considerations, we establish a criterion for a countably infinite ultrahomogeneous structure to be homomorphism-homogeneous.

scholarly journals Prime Filters and Ideals in Distributive Lattices

2013 ◽  
Vol 21 (3) ◽  
pp. 213-221 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Prime Ideal ◽  
Final Section ◽  
General Setting ◽  
Boolean Lattice ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Binary Operations ◽  
The Common ◽  
Stone Representation

Summary. The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone representation theorem [18] for Boolean or Heyting lattices. The main aim of the present article was to bridge this gap between general distributive lattices and Boolean algebras, having in mind that the more general approach will eventually replace the common proof of aforementioned articles.1 Because in Boolean algebras the notions of ultrafilters, prime filters and maximal filters coincide, we decided to construct some concrete examples of ultrafilters in nontrivial Boolean lattice. We proved also the Prime Ideal Theorem not as BPI (Boolean Prime Ideal), but in the more general setting. In the final section we present Nachbin theorems [15],[1] expressed both in terms of maximal and prime filters and as the unordered spectra of a lattice [11], [10]. This shows that if the notion of maximal and prime filters coincide in the lattice, it is Boolean.

On Endomorphism Monoids of Finite Kleene Algebras

2019 ◽  
Vol 26 (03) ◽  
pp. 507-518
Author(s):  
Jie Fang ◽  
Zhongju Sun
Keyword(s):  
Priestley Duality ◽  
Endomorphism Monoid ◽  
Endomorphism Monoids ◽  
Injective Endomorphism

An endomorphism monoid of an algebra [Formula: see text] is said to be a band if every endomorphism on [Formula: see text] is an idempotent, and it is said to be a demi-band if every non-injective endomorphism on [Formula: see text] is an idempotent. We precisely determine finite Kleene algebras whose endomorphism monoids are demi-bands and bands via Priestley duality.

scholarly journals Does the Frobenius endomorphism always generate a direct summand in the endomorphism monoids of fields of prime characteristic?

1984 ◽  
Vol 30 (3) ◽  
pp. 335-356 ◽  
Author(s):  
Péter Pröhle
Keyword(s):  
Direct Summand ◽  
Cyclic Group ◽  
Infinite Order ◽  
Main Lemma ◽  
Endomorphism Monoid ◽  
Prime Characteristic ◽  
Cancellative Monoid ◽  
Endomorphism Monoids ◽  
Technical Base ◽  
Frobenius Endomorphism

Let r be a given prime. Then a monoid M is the endomorphism monoid of a field of characteristic r if and only if either M is a finite cyclic group or M is a right cancellative monoid and M has an element of infinite order in its centre. The main lemma is the technical base of the present and other papers.

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