Demonic composition, demonic refinement and demonic union are alternatives to the usual "angelic" composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckii has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatised as the class of ordered semigroups. The proof can be used to establish that the same axiomatisation applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatisation exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatisation holds for both. We show that the isomorphism class of algebras of binary relations with the "mixed" signature of demonic composition and angelic inclusion has no finite axiomatisation. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a "mixed" signature) is finitely axiomatisable.

中文翻译：

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非确定性程序的代数：恶魔算子、命令和公理

恶魔组合、恶魔精炼和恶魔联合是对二元关系定义的通常“天使”组合、天使精炼（包含）和天使（通常）联合的替代。我们首先通过分析非确定性程序的行为来激励天使和恶魔，天使与部分正确性相关，恶魔与完全正确相关，这两种情况都来自更丰富的非确定性程序的代数模型，包括这两个方面。Zareckii 已经表明，在天使组合和包含下的二元关系代数的同构类被有限公理化为有序半群的类。该证明可用于确定相同的公理化适用于恶魔组合和细化下的二元关系，并且可以使用证明的进一步修改来合并表示天使情况下的空关系和恶魔情况下的完全关系的零元素。对于天使组合和联合的签名，已知不存在有限公理化，我们通过证明相同的公理化对两者都成立来展示恶魔组合和恶魔联合的类似结果。我们表明，具有恶魔成分和天使包含的“混合”特征的二元关系代数的同构类没有有限公理化。相比之下，我们证明了具有星座积和包含（也是“混合”签名）的部分运算的二元关系部分代数的同构类是有限公理化的。我们通过证明相同的公理化适用于两者来展示恶魔组合和恶魔联合的类似结果。我们表明，具有恶魔成分和天使包含的“混合”特征的二元关系代数的同构类没有有限公理化。相比之下，我们证明了具有星座积和包含（也是“混合”签名）的部分运算的二元关系部分代数的同构类是有限公理化的。我们通过证明相同的公理化适用于两者来展示恶魔组合和恶魔联合的类似结果。我们表明，具有恶魔成分和天使包含的“混合”特征的二元关系代数的同构类没有有限公理化。相比之下，我们证明了具有星座积和包含（也是“混合”签名）的部分运算的二元关系部分代数的同构类是有限公理化的。