### 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 [1]. 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 K_{p,o} for some p. If L ≃ 2^{k} 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 g^{2p} = id_{x}. We shall denote by X_{1},... , X_{m} the orbits of g. For each i we choose and fix a representative x_{i} ε. X_{i}. Defining c_{i} = |X_{i}|, for each i, we therefore have X_{i} = {x_{i},g(x_{i}),g^{2}(x_{i}),...,g ^{ci-1}(x_{i})}. Consider the set Λ(X;g) consisting of those mappings v:X→X that commute with g. By duality [2] 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 |
---|---|

Pages (from-to) | 99-110 |

Number of pages | 12 |

Journal | Glasgow Mathematical Journal |

Volume | 39 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Dec 1997 |

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*Glasgow Mathematical Journal*,

*39*(1), 99-110. https://doi.org/10.1017/S0017089500031967

}

*Glasgow Mathematical Journal*, vol. 39, no. 1, pp. 99-110. https://doi.org/10.1017/S0017089500031967

**Endomorphism regular Ockham algebras of finite boolean type.** / Blyth, T. S.; Silva, H. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Endomorphism regular Ockham algebras of finite boolean type

AU - Blyth, T. S.

AU - Silva, H. J.

PY - 1997/12/1

Y1 - 1997/12/1

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 [1]. 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 [2] 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 [1]. 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 [2] 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.

UR - http://www.scopus.com/inward/record.url?scp=0031495004&partnerID=8YFLogxK

U2 - 10.1017/S0017089500031967

DO - 10.1017/S0017089500031967

M3 - Article

VL - 39

SP - 99

EP - 110

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

IS - 1

ER -