scholarly journals Morita theorems for partially ordered monoids

2011 ◽  
Vol 60 (4) ◽  
pp. 221 ◽  
Author(s):  
V Laan
Keyword(s):  
Partially Ordered ◽  
Partially Ordered Monoids

Imaginary groups: lazy monoids and reversible computation

2013 ◽  
Vol 23 (5) ◽  
pp. 1002-1031 ◽  
Author(s):  
MURDOCH J. GABBAY ◽  
PETER H. KROPHOLLER
Keyword(s):  
Group Theory ◽  
Partial Ordering ◽  
Abstract Model ◽  
Sense Of Control ◽  
Reversible Computation ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Monoid Homomorphism ◽  
High Level ◽  
Partially Ordered Monoids

We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups.We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m ○ m′) ≤ f(m) ○ f(m′).Every monoid can be endowed with the discrete partial ordering (m ≤ m′ if and only if m=m′), so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of a homomorphism to this new notion of a lazy homomorphism.The computational significance of this is that a monoid is an abstract model of computation – or at least of ‘operations’ – and, similarly, a group models reversible computations/operations. With this reading, the adjunction with its injective unit gives a systematic high-level way of faithfully translating an irreversible system into a ‘lazy’ reversible one.Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add ‘undo’ (in the sense of ‘control-Z’).

Conrad’s F-condition for partially ordered monoids

2020 ◽  
Vol 24 (13) ◽  
pp. 9375-9381
Author(s):  
Pengyuan Liu ◽  
Yichuan Yang ◽  
Muhammad Zafrullah
Keyword(s):  
Partially Ordered ◽  
Partially Ordered Monoids

To the spectral theory of partially ordered sets

2018 ◽  
Vol 60 (3) ◽  
pp. 578-598
Author(s):  
Yu. L. Ershov ◽  
M. V. Schwidefsky
Keyword(s):  
Spectral Theory ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

scholarly journals Author Correction: Experimental evidence for the existence of a second partially-ordered phase of ice VI

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ryo Yamane ◽  
Kazuki Komatsu ◽  
Jun Gouchi ◽  
Yoshiya Uwatoko ◽  
Shinichi Machida ◽  
...  
Keyword(s):  
Experimental Evidence ◽  
Partially Ordered

A Correction to this paper has been published: https://doi.org/10.1038/s41467-021-22085-4

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