# Endomorphism regular Ockham algebras of finite boolean type

T. S. Blyth, H. J. Silva

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

If (L;f) is an Ockham algebra with dual space (X;g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and f is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined. An Ockham algebra (L;f) is a bounded distributive lattice L on which there is defined a dual endomorphism f. For the basic properties of Ockham algebras we refer the reader to . The most obvious example of an Ockham algebra is, of course, a boolean algebra (B; ′). In general, a boolean lattice can be made into an Ockham algebra in many different ways. Throughout what follows we shall assume that the Ockham algebra (L;f) is of finite boolean type, in the sense that L is a finite boolean lattice and f is a dual automorphism. Then (L;f) necessarily belongs to the Berman class Kp,o for some p. If L ≃ 2k then, by [1, Chapter 4], the dual space (X;g) is such that X is discretely ordered with |X| = k, and g is a permutation on X such that g2p = idx. We shall denote by X1,... , Xm the orbits of g. For each i we choose and fix a representative xi ε. Xi. Defining ci = |Xi|, for each i, we therefore have Xi = {xi,g(xi),g2(xi),...,g ci-1(xi)}. Consider the set Λ(X;g) consisting of those mappings v:X→X that commute with g. By duality  we know that (Λ(X;g), °) is a semigroup that is (anti-)isomorphic to the semigroup End((L;f), °) of Ockham endomorphisms on L. By considering Λ(X;g) we can therefore obtain properties of End(L;f). Our principal objective is to determine precisely when this semigroup is regular. For this purpose, we begin by observing the following results.

Original language English 99-110 12 Glasgow Mathematical Journal 39 1 https://doi.org/10.1017/S0017089500031967 Published - 1 Dec 1997

### Fingerprint

Ockham Algebra
Endomorphism
Semigroup
Boolean Lattice
Dual space
Cartesian product
Endomorphisms
Cyclic group
Commute
Isomorphic
Inverse Semigroup
Distributive Lattice
Boolean algebra
Bijection
Automorphism
Permutation
Duality
Choose
Orbit
Denote

### Cite this

@article{8a7d3a67cd864918bda8e24ef742e0f2,
title = "Endomorphism regular Ockham algebras of finite boolean type",
abstract = "If (L;f) is an Ockham algebra with dual space (X;g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and f is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined. An Ockham algebra (L;f) is a bounded distributive lattice L on which there is defined a dual endomorphism f. For the basic properties of Ockham algebras we refer the reader to . The most obvious example of an Ockham algebra is, of course, a boolean algebra (B; ′). In general, a boolean lattice can be made into an Ockham algebra in many different ways. Throughout what follows we shall assume that the Ockham algebra (L;f) is of finite boolean type, in the sense that L is a finite boolean lattice and f is a dual automorphism. Then (L;f) necessarily belongs to the Berman class Kp,o for some p. If L ≃ 2k then, by [1, Chapter 4], the dual space (X;g) is such that X is discretely ordered with |X| = k, and g is a permutation on X such that g2p = idx. We shall denote by X1,... , Xm the orbits of g. For each i we choose and fix a representative xi ε. Xi. Defining ci = |Xi|, for each i, we therefore have Xi = {xi,g(xi),g2(xi),...,g ci-1(xi)}. Consider the set Λ(X;g) consisting of those mappings v:X→X that commute with g. By duality  we know that (Λ(X;g), °) is a semigroup that is (anti-)isomorphic to the semigroup End((L;f), °) of Ockham endomorphisms on L. By considering Λ(X;g) we can therefore obtain properties of End(L;f). Our principal objective is to determine precisely when this semigroup is regular. For this purpose, we begin by observing the following results.",
author = "Blyth, {T. S.} and Silva, {H. J.}",
year = "1997",
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day = "1",
doi = "10.1017/S0017089500031967",
language = "English",
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In: Glasgow Mathematical Journal, Vol. 39, No. 1, 01.12.1997, p. 99-110.

Research output: Contribution to journalArticle

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AU - Blyth, T. S.

AU - Silva, H. J.

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N2 - If (L;f) is an Ockham algebra with dual space (X;g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and f is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined. An Ockham algebra (L;f) is a bounded distributive lattice L on which there is defined a dual endomorphism f. For the basic properties of Ockham algebras we refer the reader to . The most obvious example of an Ockham algebra is, of course, a boolean algebra (B; ′). In general, a boolean lattice can be made into an Ockham algebra in many different ways. Throughout what follows we shall assume that the Ockham algebra (L;f) is of finite boolean type, in the sense that L is a finite boolean lattice and f is a dual automorphism. Then (L;f) necessarily belongs to the Berman class Kp,o for some p. If L ≃ 2k then, by [1, Chapter 4], the dual space (X;g) is such that X is discretely ordered with |X| = k, and g is a permutation on X such that g2p = idx. We shall denote by X1,... , Xm the orbits of g. For each i we choose and fix a representative xi ε. Xi. Defining ci = |Xi|, for each i, we therefore have Xi = {xi,g(xi),g2(xi),...,g ci-1(xi)}. Consider the set Λ(X;g) consisting of those mappings v:X→X that commute with g. By duality  we know that (Λ(X;g), °) is a semigroup that is (anti-)isomorphic to the semigroup End((L;f), °) of Ockham endomorphisms on L. By considering Λ(X;g) we can therefore obtain properties of End(L;f). Our principal objective is to determine precisely when this semigroup is regular. For this purpose, we begin by observing the following results.

AB - If (L;f) is an Ockham algebra with dual space (X;g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and f is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined. An Ockham algebra (L;f) is a bounded distributive lattice L on which there is defined a dual endomorphism f. For the basic properties of Ockham algebras we refer the reader to . The most obvious example of an Ockham algebra is, of course, a boolean algebra (B; ′). In general, a boolean lattice can be made into an Ockham algebra in many different ways. Throughout what follows we shall assume that the Ockham algebra (L;f) is of finite boolean type, in the sense that L is a finite boolean lattice and f is a dual automorphism. Then (L;f) necessarily belongs to the Berman class Kp,o for some p. If L ≃ 2k then, by [1, Chapter 4], the dual space (X;g) is such that X is discretely ordered with |X| = k, and g is a permutation on X such that g2p = idx. We shall denote by X1,... , Xm the orbits of g. For each i we choose and fix a representative xi ε. Xi. Defining ci = |Xi|, for each i, we therefore have Xi = {xi,g(xi),g2(xi),...,g ci-1(xi)}. Consider the set Λ(X;g) consisting of those mappings v:X→X that commute with g. By duality  we know that (Λ(X;g), °) is a semigroup that is (anti-)isomorphic to the semigroup End((L;f), °) of Ockham endomorphisms on L. By considering Λ(X;g) we can therefore obtain properties of End(L;f). Our principal objective is to determine precisely when this semigroup is regular. For this purpose, we begin by observing the following results.

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